## 4 Evaluation of Stress Integration Algorithms for the e-HAH Model
### One Element Analysis
#### 4.1.1 C5T10R Loading Condition  
Figures 6 and 7 show the flow stress curves and r-value evolutions of MAT 1 and MAT 2, respectively, under the C5T10R loading path. For this analysis, three different strain increments were applied to each stress integration algorithm. The detailed values, including relative errors and CPU times, are also listed in Tables 3 and 4 for MAT 1 and MAT 2, respectively. At a first glance, all the numerical algorithms investigated seem to yield rather similar results. However, some distinctive features can be observed in the predicted flow curves and r-values. Regarding the method for the yield surface gradient calculation, both approaches based on analytical derivatives (EBM-14R/AD) and finite difference (EBM-14R/ND) predicted almost the same level of accuracy when being implemented into the fully implicit algorithm with 14 residuals. Small differences exist between the two cases in the computational cost measured by the CPU time. Indeed, the EBM-14R/ND scheme using the finite difference method takes ~8% longer CPU time than the EBM-14R/AD scheme using the analytical derivatives. Figure 5: Stress-equivalent plastic strain curve and evolution of e-HAH yield surface for MAT 2: (a) flow stress curve and (b) e-HAH yield locus for C5T10R, and (c) flow stress curve and (d) e-HAH yield locus for C5T10 CR. Figure 6: The evolution of the flow stresses and r-values of MAT 1 under C5T10R path for different stress integration algorithms with strain increments: (**a**,**b**) \(\Delta\varepsilon_{\rm xx}=5\times 10^{-3}\), (**c**,**d**) \(\Delta\varepsilon_{\rm xx}=5\times 10^{-4}\) and (**e**,**f**) \(\Delta\varepsilon_{\rm xx}=5\times 10^{-6}\). Figure 7: The evolution of the flow stresses and r-values of MAT 2 under C5T10R path for different stress integration algorithms with strain increments: (**a,b**) \(\Delta\varepsilon_{\text{xx}}=5\times 10^{-3}\), (**c,d**) \(\Delta\varepsilon_{\text{xx}}=5\times 10^{-4}\) and (**e,f**) \(\Delta\varepsilon_{\text{xx}}=5\times 10^{-6}\). The effect of the investigated algorithms on the accuracy and stability of the simulations with the e-HAH model grows considerably when the evolution of the yield surface becomes drastic within a given time increment. The flow stress curves predicted by the CPM and EBM-4R schemes show significant oscillations at the early strain range of MAT 1, when the strain increment is large, with \(\Delta\epsilon_{\text{xx}}=5\times 10^{-3}\), as shown in Figure 6a. In contrast, the two algorithms with the same strain increment are stable for MAT 2, which exhibits a lower evolution rate than MAT 1 (Figure 7a). The state variable \(g_{1}\) controls the transient rate, and Figure 8a shows its oscillations for MAT 1 when the CPM and EBM-4R schemes are used. As expected, the \(g_{1}\) value of MAT 2 exhibits a stable evolution, even for the CPM and EBM-4R algorithms. In terms of the r-value evolution, the values predicted by the CPM approach show a much lower accuracy for a strain increment of \(\Delta\epsilon_{\text{xx}}=5\times 10^{-3}\). This inaccuracy is attributed to the lack of knowledge of the flow rule as a residual during the algorithmic treatment in the CPM scheme. \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Algorithm** & \(\Delta\epsilon_{\text{xx}}\) & \(\text{Error}_{\sigma}\) (\%) & \(\text{Error}_{\text{r}}\) (\%) &
\begin{tabular}{c} **Total CPU** \\ **Time (Sec)** \\ \end{tabular} \\ \hline \multirow{3}{*}{CPM} & \(5\times 10^{-3}\) & 0.42 & 124 & 0.8 \\  & \(5\times 10^{-4}\) & 0.18 & 4.88 & 5.9 \\  & \(5\times 10^{-6}\) & 0.21 & 0.048 & 206 \\ \hline \multirow{3}{*}{EBM-4R} & \(5\times 10^{-3}\) & 0.27 & 3.43 & 0.7 \\  & \(5\times 10^{-4}\) & 0.17 & 0.19 & 6.0 \\  & \(5\times 10^{-6}\) & 0.21 & 0.022 & 202 \\ \hline \multirow{3}{*}{EBM-14R/AD} & \(5\times 10^{-3}\) & 0.49 & 0.30 & 0.6 \\  & \(5\times 10^{-4}\) & 0.25 & 0.05 & 5.7 \\  & \(5\times 10^{-6}\) & 0.22 & 0.021 & 201 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Average relative errors in the flow curves and r-values of MAT 2 under CST10R.