# Cross-Sectional Shape Optimization of Cylindrical Elastomer Spring for Sensitive Cargo Container
## 2 NFEA and Shape Optimization
### Determination of Material Coefficients for Polyurethane  
Figure 5 shows a flowchart of the process for predicting the material constants for hyperelastic behavior. First, the known material constants of the hyperelastic behavior of natural rubber were set as initial values, and the material constants were set as \(C_{10}=0.318\), \(C_{20}=-0.026\), and \(C_{30}=0.005\) for the Yeoh model equation (shown in Equation (1)). To predict the nonlinear hyperelastic behavior of polyurethane materials without additional testing, the stiffness of this polyurethane spring should be obtained in the same shape as the existing cylindrical natural rubber spring. The characteristics of this polyurethane spring were obtained from the load-displacement diagram under compression displacement control. The nonlinear material constants of polyurethane were predicted using a data-matching optimization program based on Equation (1) and NFTA. The NFEA and data-matching optimization used ABAQUS 2022 and I-SIGHT, respectively. Table 1 shows the formalization of the data-matching optimization for predicting the material constants. The objective function is to minimize the difference between the load and displacement values obtained from the nonlinear analysis results of the natural rubber spring and the compression load-displacement test result of the polyurethane spring. The iteration for the material constant prediction was as follows: \begin{table}
\begin{tabular}{c c} \hline \hline
**Design** & **Parameter** \\ \hline \multirow{2}{*}{Find} & Material constants \\  & Yeoh model (\(C_{10}\), \(C_{20}\), \(C_{30}\)) \\  & Minimize sum of squared difference \\ \multirow{2}{*}{Minimize} & \(\text{SSD}(a,b)=Min\sum_{i=1}^{n}{(a_{i}-b_{i})^{2}}\) (\(a_{i}=\) Target stiffness of \\  & polyurethane spring, \(b_{i}=\) Calibrated stiffness of polyurethane \\ \multirow{2}{*}{Target} & spring \(b_{1}=\) Stiffness of natural rubber spring) \\  & Polyurethane compressive load-displacement curve \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Formulation of data-matching optimum design. Figure 5: Flowchart of the data-matching optimization procedure. The I-SIGHT input command was saved in the I-SIGHT database as a model parameter file in the model with zrf extension file format. During each optimization iteration, the finite element model and parameter file were computed. Based on each algorithm, the result of the optimization iteration was stored in the optimization neutral form to be changed and the finite element model. From each optimization algorithm, the sum of squared difference (SSD) is iteratively performed to reduce the required condition of Table 1[29]. \[\text{SSD}(a,b)=\text{Min}\sum_{i=1}^{n}{(a_{i}-b_{i})}^{2}, \tag{2}\]  
where \(a_{i}\) is the target stiffness of the polyurethane spring, \(b_{i}\) is the calibrated stiffness of the polyurethane spring, and \(b_{1}\) is the stiffness of the natural rubber spring. Subsequently, the optimal material constant was used to compare each algorithm, and the material constant with the smallest SSD value was selected. The design data are immediately reflected by the analysis results during each stage of the optimization iteration. Figure 6 shows the compressive load-displacement diagram for the test and analysis of natural rubber and the test of polyurethane concerning the target presented in Table 1.  
Figure 7 shows the SSD history during the iterative process of the material constant prediction. This material constant prediction was obtained 100 times of iteration, which could satisfy the target in Table 1. Figure 7 shows the SSD between the predicted value of the polyurethane material constant and the test value using the number of optimization iterations based on the Hooke-Jeeves algorithm (HJA) from Figure 6. Table 2 shows the optimal prediction value at the 100 th, the smallest SSD of a total of 100 iterations from Figure 7. For reference, Table 3 shows a comparison of SSD values using the three types of optimal algorithms.