## 4 Evaluation of Stress Integration Algorithms for the e-HAH Model
### Forming Problem: S-Rail Forming and Springback
A total punch displacement of 37 mm was applied. As for the blank elements, 4264 four-node shell elements with reduced integration (S4R) were used. The simulation consisted of the holding, forming, and springback steps. Springback simulations were conducted by removing the tools after applying constraints on three specific nodes on the X'-Z' plane to prevent rigid body motion. In other words, the center node was kept fixed, one node on the Z' axis was constrained along the X' and Y' directions, and the last node on the X' axis was constrained along the Z' direction. The simulation time for each step was set to one (however, this does not have a physical meaning due to the static implicit FE algorithm. This is a relative measure for the strain increment control for different stress-integration algorithms). The reference, minimum, and maximum time increments for the forming step were set to \(5.0\times 10^{-3}\), \(1.0\times 10^{-5}\), and \(1.0\times 10^{-2}\), respectively. The sections A, B, and C of the springback are depicted in Figure 13c. Section A and C are on the Z-plane, \(\pm 115\) mm away from the center. Section B lies on the Z'-plane, which is rotated by 30\({}^{\circ}\) around the Y-axis. The springback results for section A, B, and C are presented for DP780 and STS in Figure 14. Almost no differences can be observed in the springback results obtained using the different algorithms. This may be due to the automatic time stepping built into the ABAQUS routine and to the fact that the time increment is small enough to have accurate solutions. The averaged equilibrium iteration number, averaged time increment (\(\Delta\)tavg), and relative wallclock time are listed in Table 8. The relative wallclock time is normalized with the value calculated using the CPM approach. Pronounced differences in the time increment and calculation time obtained from the different algorithms can be observed. The averaged time increment for CPM is almost half of that of the EBM algorithms. It should be considered that the tangent modulus calculated with CPM is not consistent \begin{table}
\begin{tabular}{c c c c c c c c c c} \hline \hline Material & Young’s modulus (GPa) & [PERSON]’s ratio & \multicolumn{2}{c}{K (MPa)} & \multicolumn{2}{c}{e\({}_{0}\)} & n \\ \hline DP780 & 194.0 & 0.33 & & 1295.0 & & 0.0008 & 0.142 \\ \hline STS & 223.9 & & 0.33 & & 882.6 & & 0.0056 & 0.224 \\ \hline \multicolumn{8}{c}{Yld2000-2d coefficients} \\ \hline Material & M & \(\alpha_{1}\) & \(\alpha_{2}\) & \(\alpha_{3}\) & \(\alpha_{4}\) & \(\alpha_{5}\) & \(\alpha_{6}\) & \(\alpha_{7}\) & \(\alpha_{8}\) \\ \hline DP780 & 6 & 0.946 & 1.022 & 1.015 & 1.000 & 1.011 & 0.968 & 1.009 & 1.006 \\ \hline STS & 2 & 1.0 & 1.0 & 1.0 & 1.0 & 1.0 & 1.0 & 1.0 & 1.0 \\ \hline \multicolumn{8}{c}{e-HAH model parameters} \\ \hline Material & q & k & k\({}_{1}\) & k\({}_{2}\) & k\({}_{3}\) & k\({}_{4}\) & k\({}_{5}\) & L & k\({}_{L}\) & S & k\({}_{S}\) \\ \hline DP780 & 2 & 120.0 & 150.0 & 78.5 & 0.25 & 1.0 & 0.0 & 1.0 & 0.0 & 0.8 & 75.0 \\ \hline STS & 2 & 18.7 & 580.0 & 10.0 & 0.90 & 0.88 & 240 & 1.65 & 384.0 & 1.0 & 0.0 \\ \hline \end{tabular}
\end{table} 
Table 7: Material properties and model parameters of DP780 [41] and STS [64] for e-HAH. with the Newton-Raphson method in ABAQUS, which requires a rather smaller time step, compared to the EBM algorithms. For both materials, the EBM-14R/AD and EBM-14R/ND approaches show a negligible difference in the averaged time increment and average equilibrium iteration number. In other words, the approximate computational cost for global equilibrium calculations in the FE software is similar for the two algorithms. It is thus presumed that the two algorithms have a very similar accuracy under a given time increment. For the STS case, the EBM-14R/AD and EBM-14R/ND methods show similar wallclock times, whereas these times are different for the DP780 case. Indeed, the EBM-14R/ND scheme is approximately 40% slower than the EBM-14R/AD scheme.