## 3 Mathematical Model  
A global sensitivity analysis considers the uncertainty in the input parameters and their influence on the uncertainty of the process output to rank the importance of each input parameter [12]. In this paper, we have used a variance-based global sensitivity analysis (GSA), which is commonly used to quantify the sensitivity of output to input parameters. The basic concept behind the variance-based method is to decompose the output variance among each model input. To explain the variance-based GSA, let us consider the following model:  
\[Y=f\big{(}X_{1},\ldots.X_{p}\big{)} \tag{1}\]  
where \(Y\) is the system output and \(X_{1},\ldots.X_{p}\) represents the independent input variables, which can be defined by a known probability distribution. If we compare this equation with our case, then \(Y\) can be seen as the strain value, and the inputs are the machine properties, such as laser speed and layer thickness, etc. To calculate the effect of input parameters \(X_{n}\) on the variance of \(Y\), we assume that the actual value of \(X_{n}\) is \(x_{n}^{*}\). The following conditional variance gives the change in the variance of \(Y\):  
\[O_{X_{-n}}(Y\big{|}X_{n}=x_{n}^{*}) \tag{2}\]  
In the above Equation, \(O_{X_{-n}}\) represents the conditional variance over the (p-1) input parameter space, including all the input parameters except \(x_{n}^{*}\). As the exact value of \(X_{n}\) is unknown, we will take an average over all the potential values of \(X_{n}\), which is given by:  
\[E_{X_{n}}(O_{X_{-n}}(Y\big{|}X_{n})) \tag{3}\]  
The smaller value of \(E_{X_{n}}(O_{X_{-n}}(Y\big{|}X_{n}))\) represents the greater importance of \(X_{n}\) in the variance of \(Y\). If we use the law of total variance, then we can write:  
\[O(Y)=O_{X_{n}}\big{(}E_{X_{-n}}(Y\big{|}X_{n})\big{)}+E_{X_{n}}\big{(}O_{X_{- n}}(Y\big{|}X_{n})\big{)} \tag{4}\]  
After a normalization, Equation (4) can be written as:  
\[1=\frac{O_{X_{n}}\big{(}E_{X_{-n}}(Y\big{|}X_{n})\big{)}}{O(Y)}+\frac{E_{X_{n }}\big{(}O_{X_{-n}}(Y\big{|}X_{n})\big{)}}{O(Y)} \tag{5}\]  
\[S_{n}=\frac{O_{X_{n}}\big{(}E_{X_{-n}}(Y\big{|}X_{n})\big{)}}{O(Y)} \tag{6}\]  
\(S_{n}\) represents the first-order sensitivity index for the parameter \(X_{n}\). The remaining terms in Equation (5) will help in calculating the total order index. Equation (1) is further decomposed into increasing orders of dimension, as:  
\[f\big{(}X_{1},\ldots,X_{p}\big{)}=f_{0}+\sum_{i=1}^{p}f_{i}(X_{i})+\sum_{1\leq i <j\leq p}f_{ij}\big{(}X_{i},X_{j}\big{)}+\ldots+f_{1,\ldots,p}\big{(}X_{1}, \ldots,X_{p}\big{)} \tag{7}\]In the above equation, if we assume that all the input factors are mutually independent, then there exists a simple decomposition where all the terms will be mutually orthogonal, and so the variance of the output (\(O(Y)\)) can be written as follows:  
\[O(Y)=\sum_{i=1}^{p}O_{i}+\sum_{1\leq i<j\leq p}O_{ij}+\ldots+O_{1,\ldots,p} \tag{8}\]  
where \(O_{i}\), \(O_{i,j}\),..., \(O_{1,\ldots,p}\) represents the variance of \(f_{i}\), \(f_{i,j}\),..., \(f_{1,\ldots,p}\), respectively. The first-order sensitivity index shown in Equation (6) can be obtained by using the first p term of the above decomposition.  
\[S_{i}=\frac{O_{i}}{O(Y)}=\frac{O(E(Y|X_{i}))}{O(Y)} \tag{9}\]  
Using a similar approach, we can find the higher order of sensitivity indices, such as the second-order indexes or the other higher indices.