## 3 Flow Stress Modeling
### Locally Constrained Regression Model  
A general way to model a nonlinear function, \(f\), is to approximate a model in the form \(\mathrm{y}=f(\mathbf{x};\boldsymbol{\alpha})+\varepsilon\), where \(f\) is not linear with respect to the unknown parameters \(\boldsymbol{\alpha}\), or to transform the output variable, \(\mathrm{y}\), and/or the input variable, \(\mathbf{x}\), before estimating its linear regression model, \(\mathrm{y}=\boldsymbol{\alpha}^{\mathrm{T}}\mathbf{x}+\beta+\varepsilon\). This translates into a nonlinear relationship between input and output, but the model itself is still linear in its parameters [19]. A commonly used input/output transformation is the (natural) logarithm, and the transformed model in log-log form can be expressed as [20]:  
\[\mathrm{lny}=\boldsymbol{\alpha}^{\mathrm{T}}\ln\mathbf{x}+\beta+\varepsilon \tag{14}\]  
However, there are cases wherein simply transforming the data is not adequate for more complex representations of nonlinear characteristics, thus necessitating a more general specification. Assuming \(f\) is simply a more flexible nonlinear function of \(\mathbf{x}\), a better modeling approach would be to split the global nonlinear model \(f\) into several parts and make them regionally linear, thus resulting in locally constrained regression models (LCRMs) such that [21; 22]:  
\[\mathrm{lny}=\boldsymbol{\alpha}_{\mathrm{i}}^{\mathrm{T}}\ln\mathbf{x}+\beta _{\mathrm{i}}+\varepsilon,\;\mathrm{i}=1,\cdots,\mathrm{N} \tag{15}\]  
where N is the number of local expert models and \(\varepsilon\) is the modeling error. As such, the concept of multiple modeling with switching is applied to allow the input feature domain to be naturally partitioned into multiple individual subtasks and to develop a relatively simple linear model for each partition in the region so that the set of developed models \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline
**Constant** & **5 th** & **4 th** & **3 rd** & **2 nd** & **1 st** & **0 th** \\ \hline n & 119,942.697 & \(-\)51,904.395 & 9175.732 & \(-\)809.63 & 32.563 & 4.198 \\ \(\alpha\) & 644.172 & \(-\)200.608 & 20.968 & \(-\)0.566 & \(-\)0.017 & 0.034 \\ Q & \(3.7\times 10^{9}\) & \(-1.5\times 10^{9}\) & \(2.32\times 10^{8}\) & \(-1.7\times 10^{7}\) & 398,016.905 & 151,749.123 \\ A & \(4.72\times 10^{14}\) & \(-1.9\times 10^{14}\) & \(2.71\times 10^{13}\) & \(-1.1\times 10^{12}\) & \(-5.5\times 10^{10}\) & \(4.58\times 10^{9}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Coefficient of polynomials. can be used to approximate the nonlinear properties [23]. As a result, multiple modeling schemes are very effective for modeling complex nonlinear systems. A smoother way to fit a nonlinear system is to use a piecewise quadratic or higher order trend rather than a piecewise line as described above. However, it is not recommended to fit a model higher than a quadratic to predict the stress of the aluminum alloy examined in this study because in the process of extrapolation, it can often be impractical to predict the outcome. Therefore, the least squares estimate of the parameters (\(\alpha_{\text{i}}^{\text{T}}\), \(\beta_{\text{i}}\)) is adopted due to its linearity in terms of the parameters. From this point of view, the proposed modeling scheme consists of a set of N regression models that approximate the stress properties of aluminum alloy, as well as a switching logic that selects the most optimized model based on its temperature. Figure 3 shows the overall architecture of the proposed multiple models with switching. The temperature value from 5005 aluminum alloy is provided as the input to the switching logic, which is translated into the regression model best suited to represent the stress with respect to properties such as strain and strain rate.