# Approximation Method for Stress-Strain Using Metamodel Parameter Updating
## 4 Inverse Method
### Results of the Inverse Method  
Figure 11 shows the differences in the inverse method between the typical and proposed methods based on the RMSE value. The effect of the inverse method can be observed by comparing the RMSE values in Figures 9 and 11. The polynomial model (Model 1) and exponential model (Model 2) showed relatively large errors, and the combination function (Model 3) showed the lowest RMSE value. Then, the RMSE value of the proposed method was derived to be relatively low compared to the typical method. Figure 12 shows the engineering stress-strain curves derived using the typical method. There was a difference in expressing the test data depending on the curve characteristics of the constitutive model. Since the objective function of the typical method sets only RMSE minimization, it is evident that the characteristics of the curve appear linearly, and the polynomial function (Model 1) did not approach the reference data. From these results, it is apparent that the factors were set inappropriately at the initialization stage. Inappropriate results, such as those shown in Figure 12, depend on changes in test data, MBSO algorithms, objective functions, and other factors. However, Figure 12 shows the results of using the MBSO algorithm used in this study. Figure 11: Comparison between the inverse method of the typical and proposed methods based on the RMSE values. Figure 13 shows the analysis results obtained by applying the proposed method to each constitutive model. The polynomial function (Model 1) and exponential function (Model 2) show a linear trend, similar to the results of the structural method. This result can be derived from the low RMSE value presented in Figure 11. The combination function (Model 3) of the proposed method showed the same results as the test data because the configuration model is complicated. It is observed that the result of the inversion method does not appear linear because the curve characteristics determined in the previous step are maintained. The results of the proposed methods are not always perfect, as in Model 1 and Model 2. However, despite being a suitable configuration model, the structural method did not demonstrate a good performance. Subjectively, the reason these results appear is that the inverse method attempts to curve fit both curve characteristics and curve accuracy. Figure 12: Inverse method results for typical method. Figure 13: Inverse method results for the proposed method. Figure 14 shows the RMSE and iterations for Models 1, 2, and 3. As shown in Figure 14a, the proposed method showed a relatively low RMSE value compared with the typical method. Figure 14b shows how the RMSE value changed owing to the iteration of the inverse method. Model 1 had no difference in iteration. Model 2 had more iterations of the proposed method. There were more iterations of the typical method for Model 3. This trend can be reliably compared in Figure 14c. Figure 14: Cont. Table 4 shows the parameter values derived from the inverse method. Table 5 shows the number of variables used in the inverse method, maximum number of iterations until convergence, number of SE simulations required by the design of the experiment, sum of FE simulations until convergence, and convergence result represented by RMSE. Table 5 has the total number of interpretations based on Equation (8). The proposed method saved 1.4 times more resources in Model 1 and 5.3 times more in Model 3.