## 3 ANN Flow Law Implementation in FE Software
### Numerical Simulations and Benchmarks Tests
Figure 10 shows the evolution of the maximum radius \(r\) of the cylinder (measured at the middle of the sample height) as a function of the vertical displacement of the top of the cylinder. This figure shows a slight difference of the four models during the numerical simulation. In the rest of this section, the four models will be referred to as \(M_{1234}\). The final value therefore differs from \(r=5.870\) mm for \(M_{1}\) to \(r=5.917\) mm for \(M_{34}\). Table 2 gathers the results allowing the comparison of the four identified flow laws. It appears from the study of this table that the modification of the number of neurons in the hidden layers has an influence on the computation time \(t\), which increases with the complexity of the network structure as expected and varies within the range from 886 s to 1263 s approximately since this information is hard to capture from a commercial software that does not contain accurate CPU time reports as the Abaqus software. It is obvious from this table that all models do not give exactly the same results. The computing time increases from \(M_{1}\) to \(M_{4}\), proof that the increase in complexity of the ANN has an influence on the global computation time \(t\). \begin{table}
\begin{tabular}{l c c c c c c} \hline \hline
**Model** & **ANN** & \(N_{inc}\) & \(t\) & \(r\) & \(\varepsilon^{p}\) & \(T\) & \(\sigma\) \\  & & **(s)** & **(mm)** & & **(\({}^{\circ}\)C)** & **(MPa)** \\ \hline \(M_{1}\) & 3-7-4-1 & 1,367,147 & 886 & 5.870 & 0.891 & 794.74 & 161.95 \\ \(M_{2}\) & 3-9-4-1 & 1,405,471 & 941 & 5.895 & 0.927 & 798.29 & 178.78 \\ \(M_{3}\) & 3-9-7-1 & 1,408,680 & 1023 & 5.917 & 0.965 & 798.76 & 164.25 \\ \(M_{4}\) & 3-15-7-1 &The results, both from the point of view of the dimensional characteristics of the sample (maximum radius \(r\)), or the internal fields, such as the temperatures \(T\) and the plastic strains \(\varepsilon^{p}\) and equivalent stresses \(\sigma\) in the center of the sample, are more or less the same for all \(M_{1234}\) flow models. To have a better analysis of the difference between models \(M_{1234}\), Figure 11 shows the evolution of the equivalent von Mises stress \(\sigma\) for the element in the center of the cylinder during the compression. As shown in this figure, the \(M_{1234}\) models give equivalent results with more or less the same value of the equivalent stress \(\sigma\) at the end of the cylinder compression as shown in Table 2. The appearance of the curve for the \(M_{3}\) and \(M_{4}\) models is very oscillating in the first 1/3 of the graph. This is probably related to the fact that the \(M_{34}\) models are over-fitted and cannot serve as a universal approximator for the flow stress. To verify this assumption, Figure 12 shows a plot of the predicted flow stress \(\sigma^{y}\) using the 3-15-7-1 ANN model as a function of the plastic strain \(\varepsilon^{p}\) and the plastic strain rate \(\dot{\varepsilon}\) for a fixed temperature \(T=750\)\({}^{\circ}\)C.  
Figure 11: Evolution of the equivalent von Mises stress \(\sigma\) of the specimen during the compression process using the four ANN flow laws. Figure 12: Predicted flow stress \(\sigma^{y}\) as a function of \((\varepsilon^{p},\dot{\varepsilon})\) for a fixed temperature \(T=750\)\({}^{\circ}\)C using the 3-15-7-1 model. This figure shows that when \(\dot{\epsilon}\) varies between 0.001 s\({}^{-1}\) and 0.01 s\({}^{-1}\), and for a value of \(\epsilon^{p}<0.3\), the representative curves of \(\sigma^{y}\) 'cross' demonstrate the poor interpolation of the flow stresses \(\sigma^{y}\) when the strain rate varies (there is a zone in black dashed lines above the blue line). Thus, for a strain rate between 0.001 s\({}^{-1}\) and 0.01 s\({}^{-1}\), and for a plastic strain less than 0.3, the flow stress \(\sigma^{y}\) is greater than the value of \(\sigma^{y}\) calculated for the same plastic strain and \(\dot{\epsilon}=0.001\) s\({}^{-1}\), which is physically not admissible for the behavior of GCr15 material.