## 4 Finite Element Analysis
### FE Mesh Convergence Study  
This section is devoted to a convergence study carried out on the FE mesh to assess how the mesh would affect the results and particularly to determine the optimal mesh density. The chassis is considered a plate on which shell elements can be employed to analyse the problem. Different shell element types were studied: quadratic elements with/without reduced integration for four and eight nodes identified as S4R, S4, and S8R.  
To achieve the optimal mesh, the model presented in Figure 8 was analysed. Regarding the boundary condition, a vertical displacement as \(V(x,y)=10\ (\mathrm{mm})\) was imposed at Node 1. The numerical simulations have been performed on different FE mesh types with an element size of \(200\ (\mathrm{mm})\), which is reported in Table 4. However, Table 5 presents a summary of the obtained displacement along the \(z\)-direction. As a conclusion, different element types do not have a significant impact on the results. Based on the findings of the FE convergence investigation, it can be concluded that the S4R element type would be the best option for ensuring solution correctness and minimizing computing costs.  
Maintaining the same boundary conditions mentioned before and considering the different FE meshes reported in Table 6, the problem was numerically solved, and the displacement variation over \(z\)-direction was obtained for all nodes as reported in Table 7.  
Owing to the results obtained from various FE densities, it can be inferred that medium mesh density produced accurate results while minimizing computational costs and increasing convergence.  
\begin{table}
\begin{tabular}{c c c c} \hline \hline \multirow{2}{*}{**FE Mesh**} & \multirow{2}{*}{**Coarse**} & \multirow{2}{*}{**Medium**} & \multirow{2}{*}{**Fine**} \\ \hline Element size \((\mathrm{mm}\ )\) & 400 & 200 & 100 \\ Number of elements & 217 & 840 & 3213 \\ Number of nodes & 256 & 915 & 3360 \\ \hline \hline \end{tabular}
\end{table}
Table 6: FE mesh properties for the element type of S4R.  
\begin{table}
\begin{tabular}{c c c c} \hline \hline \multirow{2}{*}{**Displacement Imposition at Node 1**} & \multirow{2}{*}{**Corner**} & \multicolumn{3}{c}{\(\boldsymbol{u_{z}(\mathrm{mm})}\)} \\ \cline{3-4}  & & **S4R** & **S4** & **S8R** \\ \hline Number of elements & 1 & 10.00 & 10.00 & 10.00 \\ Number of nodes & 915 & 915 & 2669 \\ \hline \hline \end{tabular}
\end{table}
Table 4: FE mesh properties for different element types.  
\begin{table}
\begin{tabular}{c c c c} \hline \hline \multirow{2}{*}{**Displacement Imposition at Node 1**} & \multirow{2}{*}{**Corner**} & \multicolumn{3}{c}{\(\boldsymbol{u_{z}(\mathrm{mm})}\)} \\ \cline{3-4}  & & **Coarse** & **Medium** & **Fine** \\ \hline \multirow{4}{*}{\(10\ (\mathrm{mm})\)} & 1 & 10.00 & 10.00 & 10.00 \\  & 2 & 3.32 & 3.33 & 3.33 \\ \cline{1-1}  & 3 & \(-3.32\) & \(-3.33\) & \(-3.33\) \\ \cline{1-1}  & 4 & 3.34 & 3.33 & 3.33 \\ \hline \hline \end{tabular}
\end{table}
Table 7: Displacement along \(z\)-direction for different mesh densities of S4R.  
\begin{table}
\begin{tabular}{c c c c} \hline \hline \multirow{2}{*}{**Element Type**} & \multirow{2}{*}{**S4R**} & \multirow{2}{*}{**S4**} & \multirow{2}{*}{**S8R**} \\ \hline Number of elements & 840 & 840 & 840 \\ Number of nodes & 915 & 915 & 2669 \\ \hline \hline \end{tabular}
\end{table}
Table 5: Displacement along \(z\)-direction for different FE types.