## 4 Semi-Automated Procedure to Estimate the NKH Model Parameters for Soils and Rocks
### Semi-Automated Procedure to Estimate the NKH Model Parameters
Then, each pair of kinematic hardening parameters, including \(\mathrm{C_{1}}\)-\(\gamma_{1}\), \(\mathrm{C_{2}}\)-\(\gamma_{2}\), and \(\mathrm{C_{3}}\)-\(\gamma_{3}\), respectively, were modified with a reducing changing rate to fit each part of the \(\mathrm{G/G_{max}}\) and/or damping curves. The changing rate, which could be manually input into MATLAB, indicates the change in the percent of the parameter (i.e., 20%, 10%, and 5%), and the degree of change should depend on the MAE value. The weighting factor  
Figure 7: Flowchart for estimating NKH model parameters. (\(w\)) was used to estimate the error considering both G/Gmax and damping with MAEG and MAED, respectively, as follows:  
\[\text{MAE}=w\text{-}\text{MAEG}+(1-w)\text{-}\text{MAED} \tag{7}\]  
Note that in fitting to the damping curve, the new target damping was considered by subtracting the damping (i.e., damping obtained from the RCT) to the initial damping value, which was defined as damping at the threshold strain. New back-stress parameters for each segment of G/Gmax or damping curves could be obtained with the lowest MAE value at that segment, following the method proposed by [PERSON] and [PERSON] (2018). The convergence of the procedure could be reached if the MAE value was stable, with  
Figure 8: G/Gmax and damping curves with initial parameters of the NKH model for (**a**) silica sand, (**b**) mixture soil, and (**c**) Toyoura sand. slight modification in the rate of change. Figure 9a-c show the results of the G/Gmax and damping curves for the three soils with parameters obtained by the proposed procedure with \(w\) factors of 1, which means that only G/Gmax was considered during the modification process. Acceptable results were obtained in G/Gmax compared between the model and target curves, which show considerable improvement compared to the initial parameters in Figure 8a-c. By performing the algorithm in the MATLAB compiler, the results of the parameters were obtained in a very short time, compared to the manual trial-and-error method mentioned above. The results for the damping improved, even though the underrated damping was seen at small shear strain and damping was overestimated at large shear strains. Figure 9d compares the back-stress curves of Toyoura sand for three cases, including the SNKH model, the NKH model with parameters obtained from the semi-automated procedure (NKH_G), and from modifying the SNKH curves through trial-and-error (NKH_Trial). The back-stress value in large plastic strain value obtained by the semi-automated method is reasonable compared to those of the trial-and-error method and the SNKH model. The same results were also obtained for the silica sand and mixture soil. It could be concluded that the NKH model and its parameters obtained by the algorithm match the requirement for the shear strength of the material proposed by [PERSON] and [PERSON] [5]. Figure 10 shows the results of the shear modulus degradation and damping curves for the three soils with parameters obtained by the semi-automated procedure with a \(w\) factor of 0.95, which means that both the G/Gmax and damping were considered. The results for damping were much improved for all materials compared to those in Figure 9a-c with acceptable shear modulus degradation curves. The value of the weighting factor could  
Figure 9: G/Gmax and damping curves with parameters of the NKH model (weighting factor, \(w=1\)) for (**a**) silica sand, (**b**) mixture soil, and (**c**) Toyoura sand, and (**d**) back-stress curve for Toyoura sand. be increased to improve the accuracy of the model in simulating the shear modulus of the material.