## 2 Materials and Methods
### Van Rollover Test and Model
#### 2.1.2 Establishment and Verification of the Van Rollover Model  
The causes and conditions of rollover accidents vary greatly, and due to the high cost and lengthy cycle of platform rollover tests, this study developed a finite element model of a van for simulation analysis. The finite element model of the van is depicted in Figure 3. It comprised 743 components, including 993,356 elements and 1,042,370 nodes. According to FMVSS208 regulations, the van's rolling condition platforms were established and the following parameters were set.  
1. Two rigid floors and one rigid baffle were established. The tire material was set to Ogden hyper-elastic material. The car model was tilted around its vertical axis by 23\({}^{\circ}\), and the baffle height was set to 0.1016 m. The lowest point of the tire was 0.2286 m away from the rigid floor, and the tire was in close contact with the rigid baffle.
2. Load setting: The velocity along the negative direction of the Y-axis was 13.33 m/s, and the gravitational acceleration was set to 9.8 m/s\({}^{2}\).
3. Contact settings: In order to be closer to the experimental conditions, the tire was in contact with the stop surface and with the rigid ground on one surface; the vehicle body (except for the tire) was in contact with the rigid ground on two surfaces; and the vehicle was in self-contact.
4. Friction coefficient setting: The friction coefficient between the tire and the baffle was set to 0.2, the friction coefficient between the tire and the rigid ground 1 was 0.75, and the friction coefficient between the vehicle body (excluding the tire) and the rigid ground 2 was 0.4.  
LS-DYNA was utilized to analyze the finite element model of the van, and the model was validated from three perspectives: key motion state benchmarking, acceleration benchmarking, and roof load displacement curve benchmarking.  
The instant at which the test vehicle and the test bench began to move relative to each other was defined as t0 (measured in ms). The following key movement states were determined: rolling 1/4-week, 1/2-week, 3/4-week, 1-week, 5/4-week, 3/2-week, 7/4-week, and 2-weeks. The time errors for each movement state were as follows: 0.4%, \(-\)0.5%, \(-\)0.8%, 0.4%, 0.5%, 0.4%, 1.7%, 2.2%, and 2.4%, respectively. Figure 4 illustrates a comparison of the key motion states of the test (left) and simulation (right), which reveals that the motion attitude deviation between the test and simulation results was minimal.  
Figure 3: FE model of van.  
In Figure 5, a comparison of the acceleration rates at the vehicle centroid in the test and in the simulation reveals that the trend of the three-way acceleration curve in both cases was essentially consistent, with peak values at each moment of ground impact. The level of fit was high, and although the peak values differed slightly, they fell within an acceptable range of \(\pm\)5%.  
Figure 6 depicts a comparison between the FE model calculation results of the van under top pressure test conditions and the corresponding test results. It can be observed that the FE model load displacement curve of the van aligns well with the test results, indicating the high reliability of the model. These three test benchmarks validate the efficacy of the FE model of the van in research on crashworthiness and weight optimization.  
Figure 4: Comparison of key motion states: (**a**) tire touchdown; (**b**) \(1/4\)-week; (**c**) \(1/2\)-week; (**d**) \(3/4\)-week; (**e**) \(1\)-week; (**f**) \(5/4\)-week; (**g**) \(3/2\)-week; (**h**) \(7/4\)-week; (**i**) \(2\)-week.  
Figure 5: _Cont._