# Approximation Method for Stress-Strain Using Metamodel Parameter Updating
## 4 Inverse Method
### Inverse Method Using Metamodel-Based Simulation Optimization  
Figure 10 shows the relationship between the pre-processing step and the inverse method step. Equation (5) shows the most complex form (Model 3) of the configuration model by pre-process. Equation (6) shows Model 3 with Matlab CFT with the scale factor (\(s_{1}\)) applied. Equation (7) shows an application example of the scale factor (\(s_{2}\)) applied to the true strain value.  
\[\sigma_{t}=(\alpha)\{\sigma_{y}+K(\varepsilon_{y}+\varepsilon_{t})^{n}\}+(1- \alpha)\{\sigma_{s}-\big{(}\sigma_{s}-\sigma_{y}\big{)}exp(-b\varepsilon_{t}{}^ {p})\} \tag{5}\]  
\[\sigma_{t}=s_{1}\big{[}(\alpha)\{\sigma_{y}+K(\varepsilon_{y}+\varepsilon_{t}) ^{n}\}+(1-\alpha)\{\sigma_{s}-\big{(}\sigma_{s}-\sigma_{y}\big{)}exp(-b \varepsilon_{t}{}^{p})\}\big{]} \tag{6}\]  
\[\varepsilon_{t}=s_{2}[\ln(1+\varepsilon_{e})] \tag{7}\]  
Figure 9: Comparison between RMSE values of FE simulation results and test results.  
Figure 10: _Cont._  
As shown in Figure 10, constitutive Models 1, 2, and 3 were used for both the typical method and the proposed method.  
In the pre-processing step, a tensile test was performed, and the configuration model was initialized via the load-displacement curve. In addition, the range of parameters was updated based on the RMSE value in the inverse method process. The D-Optimal method was used for the experiment design, which is advantageous for generating a regression equation-based response surface model [51,52]. The FE simulation was performed repeatedly by updating the parameters using the design of the experiment. In this process, the quantitative error of the FE simulation was calculated using RMSE, and the correlation between the parameter changes and RMSE was approximated by the response surface, which is composed of a linear equation. When an error occurred, it was composed of a quadratic polynomial equation. This nonlinearity of the response surface influenced the design of the experiment and the number of FE analysis iterations performed.  
Figure 10a shows the flow of a typical method. In the pre-processing stage, only the parameters were initialized using the ASTM E646 standard, and for the inverse method, the parameter set X was used for the design of the experiment as they were. Figure 10b shows the flow of the proposed method. During the pre-processing, the parameters were initialized using the ASTM E646 standard and MATLAB CFT, and the scale parameters of the stress and strain axes for the true stress-strain curve were used as the set of inverse method parameters X.  
\[v=\{q(k_{max}-1)\}+1 \tag{8}\]  
The total iteration count (\(v\)) of the FE simulation required by the inverse method can be calculated using Equation (8) ([PERSON] et al., 2005). Here, \(k\) denotes the iteration,  
Figure 10: Inverse method procedure: (**a**) Typical method (metamodel-based simulation optimization with one-time initialization of variables); (**b**) Proposed method (metamodel-based simulation optimization using CFT); (**c**) Concept diagram of inverse method procedure.  
and \(q\) is the required iteration count of the FE simulation per iteration. The value of \(q\) may vary according to the number of parameters, the curvature of the response surface, and the theory of the experiment design. In the last iteration of the inverse method, only one FE simulation was performed for verification. A genetic algorithm was used to analyze the response surface model and was used to derive the lowest RMSE within the range of the response surface and the parameter value that it satisfied. Convergence was determined through changes in the RMSE values and parameters. If the RMSE values from each FE simulation were larger than 0.01, the parameter range was adjusted around the optimal point, and the experimental plan was recreated. The inverse method was set to end when the average change in RMSE was less than 0.01%, the amount of average change in the parameter was less than 0.01, and the number of iterations was 50.