## 2 Materials and Methods
### Identification of the Constants of GTN Model
#### 2.3.5 Kriging Method  
In this model the prediction function \(y(x)\) is a linear combination of the main function \(f(x)\) and random function \(Z(x)\)[24]:  
\[y(x)=f(x)+Z(x), \tag{24}\]  
The function \(f(x)\) is usually determined by a polynomial or root base function. \(Z(x)\) is a Gaussian random function with zero average, non-zero variance \(\sigma^{2}\) and also covariance defined as:  
\[cov[Z(x),Z(\hat{x})]=\sigma^{2}R(x,\hat{x}), \tag{25}\]  
where, \(R(x,\hat{x})\) is a correlation function which is just dependent to two vectors \(x\) and \(\hat{x}\). \[R(x,\hat{x})=\exp\Bigl{(}-\sum_{k=1}^{m}\theta_{k}|x_{k}-\hat{x}_{k}|^{c_{k}} \Bigr{)}, \tag{26}\]  
In which \(\theta_{k}\) and \(c_{k}\) are the Kriging unknown coefficients in the range of \(0<\theta_{k}<\infty\) and \(1<c_{k}\leq 2\). The number of these coefficients are equal to the number of design parameters. The values of these parameters determine the effect of each design parameter r r non the objective function. The relation Equation (24) can be rewritten in matrix form as:  
\[y(x)=\beta_{0}+r^{T}(x)R^{-1}(y_{s}-\beta_{0}I), \tag{27}\]  
where, \(\beta_{0}\) is the least square estimation defined as follows:  
\[\beta_{0}=\Bigl{(}I^{T}R^{-1}I\Bigr{)}I^{T}R^{-1}y_{s}, \tag{28}\] \begin{table}
\begin{tabular}{c c c c c c c} \hline \(f_{c}\) & \(f_{0}\) & \(f_{n}\) & \(f_{f}\) & **Error \%** \\ \hline
0.148 & 0.013 & 0.046 & 0.246 & 0.72 \\ \hline \end{tabular}
\end{table} 
Table 2: The constants of GTN damage model obtained using polynomial regression method. \(I\) is the unit vector and \(y_{s}\) is the function output in initial sample points. \(r\) and \(R\) are correlation vector and correlation matrix, respectively,  
\[R=\left[ \begin{array}{cccc}R\Big{(}x^{(1)},x^{(1)}\Big{)}&R\Big{(}x^{(1)},x^{( 2)}\Big{)}&\ldots&R\Big{(}x^{(1)},x^{(n)}\Big{)}\\ R\Big{(}x^{(2)},x^{(1)}\Big{)}&R\Big{(}x^{(2)},x^{(2)}\Big{)}&\ldots&R\Big{(}x^{( 2)},x^{(n)}\Big{)}\\ &\ldots&\\ R\Big{(}x^{(n)},x^{(1)}\Big{)}&R\Big{(}x^{(n)},x^{(2)}\Big{)}&\ldots&R\Big{(}x^ {(n)},x^{(n)}\Big{)}\end{array} \right],r=\left[ \begin{array}{c}R\Big{(}x^{(1) },x\Big{)}\\ R\Big{(}x^{(2)},x\Big{)}\\ \ldots\\ R\Big{(}x^{(n)},x\Big{)}\end{array} \right], \tag{29}\]  
where, \(x\) and \(x(i)\) are design parameters vectors for which objective functions are unknown and known respectively. The variance can be calculated as,  
\[\sigma^{2}(\beta_{0},\theta,c)=\bigg{(}\frac{1}{n}(y_{s}-\beta_{0}1)^{T}R^{-1 }(y_{s}-\beta_{0}1)\bigg{)}, \tag{30}\]  
Assuming the Gaussian distribution of the samples the likelihood function would be:  
\[L\Big{(}\beta_{0},\sigma^{2},\theta,c\Big{)}=\frac{1}{\sqrt{2\pi(\sigma^{2}) ^{n}|R|}}\exp\Bigg{\{}-\frac{1}{2}\frac{(y_{s}-\beta_{0}1)^{T}R^{-1}(y_{s}- \beta_{0}1)}{\sigma^{2}}\Bigg{\}}, \tag{31}\]  
The later equation can be written as:  
\[MLE(\theta,c)=-nln\sigma^{2}(\theta)-lnR(\theta), \tag{32}\]  
This is the equation that should be optimized in order to obtain the Kriging Unknown parameters. For simplicity, we considered \(c_{k}=2\). Equation (32) is optimized by genetic algorithm. In the polynomial regression method, the number of initial samples is determined according to the number of optimization problem parameters. For example, in this investigation there are four parameters for which the linear polynomial function has 15 constants. Therefore, 15 simulation samples are required to determine the polynomial constants. Oppositely, Kriging function does not require specific number of initial samples. At the beginning, the same 15 samples used for polynomial regression method formation were used for constructing the Kriging function. Therefore, Kriging parameters were obtained by optimizing the relation Equation (32) using genetic algorithm. Then by optimizing the Kriging function again with genetic algorithm, the constants of GTN damage were obtained. The identified constants were then used in accomplishing the simulation.