# Structural Optimization of Vented Brake Rotors with a Fully Parameterized Model
## 2 Research Methods
### Convergence Test  
Before performing further finite element analyses and optimizations, a convergence test was executed to ensure the credibility of the finite element model. Gradually reducing the element sizes, remeshing the solid model with higher-order 3D elements, and performing multiple modal analyses on the baseline model of the rotor under a free-free boundary condition revealed its first 40 natural frequencies and mode shapes, excluding the rigid body modes. A total of 12 finite analyses were performed, with element sizes reducing from 10 mm to 1.7 mm. Table 3 gives the element sizes and the resulting numbers of nodes in the finite element models, with more than 1.25 million nodes in the model of element size 1.7 mm. All 40 natural frequencies appeared to converge to certain values as the number of nodes increased. Figure 9a,b illustrate, respectively, the frequency converging trends for the 2 ND mode and 1T mode.  
Since the subsequent optimization tasks in this research require a great number of repeated finite element calculations, an accurate finite element model with a proper mesh size should be established before optimization to reduce the computational burden. A relative error in computed frequency for a particular mode (_i_) and with element size (_s_) is defined as  
\[\epsilon_{i,s}=\frac{f_{i,s}-f_{i,1,7}}{f_{i,1,7}} \tag{1}\]  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Young’s Modulus (GPa)** & **Poison’s Ratio** & **Density** & **Thermal Exp.** & **Thermal Conduct.** & **Specific Heat** \\  & **(kg/m\({}^{3}\))** & **Coef. (I/\({}^{5}\)C)** & **(W/(m\({}^{5}\)C))** & **(J/(kg\({}^{5}\)C))** \\ \hline
128.75 & 0.28 & 7220 & \(1.1\times 10^{-5}\) & 55 & 500 \\ \hline \hline \end{tabular}
\end{table}
Table 2: Material properties used in the finite element models.  
\begin{table}
\begin{tabular}{c c c} \hline \hline
**Element Size (mm)** & **No. of Nodes** & **Max. Relative Error** \\ \hline
10 & 23,953 & 5.49\% (35 th mode) \\
8 & 30,170 & 5.40\% (35 th mode) \\
6 & 47,723 & 4.45\% (35 th mode) \\
5 & 69,025 & 3.89\% (35 th mode) \\
4 & 135,872 & 1.94\% (35 th mode) \\
3 & 281,539 & 1.02\% (35 th mode) \\
2.5 & 429,193 & 0.57\% (35 th mode) \\
2.2 & 535,311 & 0.44\% (35 th mode) \\
2.1 & 662,995 & 0.33\% (35 th mode) \\
2 & 790,894 & 0.19\% (35 th mode) \\
1.8 & 1,096,105 & 0.04\% (8 th mode) \\
1.7 & 1,250,497 & - \\ \hline \hline \end{tabular}
\end{table}
Table 3: Element sizes and numbers of nodes in the model.  
Figure 9: Frequency convergence history: (**a**) 2 ND mode; (**b**) 1T mode.  
where \(f_{i,s}\) is the natural frequency for mode \(i\) of the finite element model with element size \(s\), and \(f_{i,1,7}\) is computed from the finite element model with the element size of 1.7 mm. In Table 3, the maximum relative errors and the corresponding modes for element sizes ranging from 10 mm to 1.7 mm are also listed. The smallest model (with the fewest nodes) having relative errors, defined in Equation (1), of less than 1% in all 40 natural frequencies was selected for the subsequent optimization problem. The resulting finite element model, having an element size of 2.5 mm, consisted of 275,226 higher-order elements and 429,193 nodes (shown in Figure 7) and produced frequency errors of less than 0.6% for all 40 modes. The 2 ND through 7 ND modes and the 1T mode predicted by the selected model are shown in Figure 8, in which, the 1T mode, for example, is the 30 th mode, with a frequency of 9701 Hz, and 7 ND mode is at 10,317 Hz (the 34 th mode).