# Cross-Sectional Shape Optimization of Cylindrical Elastomer Spring for Sensitive Cargo Container
## 2 NFEA and Shape Optimization
### Cross-Section Shape Optimization  
The design requirements of this cylindrical polyurethane spring are to satisfy the target stiffness listed in Table 4 and reduce the MPS [2]. Figure 10: MPS distribution and histories of the original shape (shape 6 in Figure 4): (**a**) NFEA results; (**b**) comparison of MPS history between natural rubber and polyurethane. Figure 9: MPS and CPU time vs. mesh size. From a conservation design perspective, the design target stiffness (k) was then derived to prevent vibration and impact set at 0.25 to 2.5 Hz, i.e., natural frequency (f) based on a load cargo load (m) of 0.2 to 20 tons. Here, the resonance frequency f = \(1/2\pi\times\sqrt{k/m}\) was derived by assuming six springs were installed on one pallet. Therefore, an optimal design is performed. Table 5 summarizes the formulation of the optimal design of the cross-section shape of this polyurethane spring based on the design parameters listed in Figure 11.  
The maximum vibration frequency of the container was set to 5 and 40 Hz during truck and freight car operation, respectively, based on the results of previous studies [30]. Figure 12 shows the flow of the optimal design process for the cross-section shape of the spring. First, a three-dimensional model drawing was prepared using the CAD program CATIA V5. Subsequently, preprocessing and nonlinear analysis were performed using the NFEA program ABAQUS (version 2022). Based on the results shown in Figure 10, the initial conditions of the optimization stage set as the design area and variables defined in Figure 11 and Table 5, and the ABAQUS odd file was designated as the objective function result file. The optimal design algorithms were three general-purpose optimization algorithms (i.e., OLH, HJA, and MIGA) provided by I-SIGHT. The iteration for the optimal design was as follows:  
I-SIGHT input commands were stored in the I-SIGHT database as a model parameter file in the model zrf format. During the optimization iteration, the finite element model \begin{table}
\begin{tabular}{c c} \hline \hline
**Technical Requirement** & **Value** \\ \hline Maximum compressive displacement under 3.4 kN & 20 mm \\ Maximum principal strain & Minimum value \\ Natural frequency & \(\leq\)5 Hz \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Technical requirements of polyurethane spring. Figure 11: Design parameters for polyurethane spring. and parameter file were computed simultaneously. Based on each algorithm, the result of the optimization iteration was stored in the optimized neutral form, and the finite element model was changed. Based on each optimal algorithm, the MPS was reduced iteratively while satisfying the constraints listed in Table 5.  
The optimal cross-section shapes were compared for each algorithm, and the cross-sectional shape with the smallest MPS value was selected. These design data were immediately reflected in the analysis results during each stage of the optimization iteration. Figure 13 shows the history of the MPS in the iterative process of shape optimization. The optimized shape was obtained at 447 th, the smallest MPS of a total of 500 iterations from Figure 13. Required design targets listed in Tables 4 and 5, i.e., \"20 mm displacement under a compressive load of 3.4 kN\" and \"minimum value of MPS,\" were satisfied. Figure 13 shows the optimal design result of the polyurethane spring using the MIGA presented in Figure 12.  
Figure 12: Flowchart of the iterative optimization procedure. Figure 13: Optimization histories using MIGA: (**a**) optimization history; (**b**) detailed view of optimization history. Figure 14a shows the distribution of the MPS for the optimal design of this spring, and Figure 14b shows the history of the MPS at the weakest point in the original drawing and the optimal design section.