## Material and Methods
### Overview of GMMH  
Group method of data handling (GMDH) contains a set of neurons created by linking different pairs through a second-degree polynomial. The neural network combines the second-degree polynomials obtained from all the neurons and describes an approximate \(f\) function with an \(\hat{y}\) output for a set of \(X=(x_{1},x_{2},x_{3},\ldots,x_{n})\) inputs with the minimum error in comparison with the real output, \(y\)  
Figure 4: Boundary condition of the simulated model using the ANSYS-CFX.  
Figure 3: Gridding the plan and the cross-section of the simulated model using the ANSYS-CFX.  
The actual results for \(M\) experimental data with \(n\) inputs and one output is therefore shown as Equation (1) [21; 26; 27; 29; 30]:  
\[y_{i}=f(x_{i1},x_{i2},x_{i3},\ldots x_{in})\hskip 56.905512 pt(i=1,2,3,\ldots,M) \tag{1}\]  
We aim to obtain a network which could predict the value of \(\hat{y}\) output for each \(X\) input axis based on the Equation (2):  
\[\hat{y}_{i}=\hat{f}(x_{i1},x_{i2},x_{i3},\ldots x_{in})\hskip 56.905512 pt(i=1,2,3, \ldots,M) \tag{2}\]  
It must also minimize the error of the predicted and actual values as shown below in addition to predicting the above equation:  
\[MSE=\frac{1}{M}\sum_{i=1}^{M}\left(\hat{y}_{i}-y_{i}\right)^{2}\to Min \tag{3}\]  
The general form of the link between the input and output variables can be presented as below through utilizing a polynomial function,  
\[y=a_{0}+\sum_{i=1}^{n}a_{i}x_{i}+\sum_{i=1}^{n}\sum_{j=1}^{n}a_{ij}x_{i}x_{j}+ \sum_{i=1}^{n}\sum_{j=1}^{n}\sum_{k=1}^{n}a_{ijk}x_{i}x_{j}x_{k}+\ldots \tag{4}\]  
The second degree of the above-mentioned equations, which will be used in this research, can be written as below [21; 27; 29; 30]:  
\[\hat{y}=G\Big{(}x_{i},x_{j}\Big{)}=a_{0}+a_{1}x_{i}+a_{2}x_{j}+a_{3}x_{i}^{2}+ a_{4}x_{j}^{2}+a_{5}x_{i}x_{j} \tag{5}\]  
The \(a_{i}\) unknown coefficients are obtained by regression methods in such a manner which the differences between the real output '\(y\)' and the calculated '\(\hat{y}\)' values are minimized for each pair of input variable \(x_{i}\) and \(x_{j}\). A set of polynomials is made by considering the above-mentioned equation and all their unknown coefficients are obtained through considering the least square approach [21; 27; 29; 30]. The coefficients of the equations of each neuron for minimizing its total error, to efficiently match the inputs with all the pairs of input-output sets, are obtained for each \(G_{i}\) function (each generated neuron).  
\[E=\frac{1}{M}\sum_{i=1}^{M}\left(y_{i}-G_{i}\right)^{2}\to Min \tag{6}\]  
In primary methods of GMDH, all paired combinations (neurons) are generated from the input and the unknown coefficients of all neurons are obtained by using the least square. Therefore \(\left(\begin{array}{c}\text{n}\\ 2\end{array}\right)=\frac{\text{n}(\text{n}-1)}{2}\) neurons are generated in the second layer which could be expressed as shown below [21; 29; 30]:  
\[\Big{(}\Big{(}y_{i},x_{ip},x_{iq}\Big{)}\Big{|}(i=1,2,\ldots,M)\&\ p,qq\in(1,2,\ldots,M)\Big{\}} \tag{7}\]  
We will use the second degree of the function in Equation (5) for all M columns of three. The equations could be expressed in a matrix as follows [29; 30]:  
\[Aa=Y \tag{8}\]  
where \(A\) is the axis of the unknown coefficients of the second-degree equation indicated in Equation (5), meaning:  
\[a=\{a_{0},a_{1},\ldots,a_{5}\} \tag{9}\]\[Y=\{y_{1},y_{2},y_{3},\ldots,y_{M}\} \tag{10}\]  
\[A=\left[\begin{array}{ccccc}1&x_{1p}&x_{1p}^{2}&x_{1q}^{2}&x_{1p}x_{1q}\\ 1&x_{2p}&x_{2p}^{2}&x_{2q}^{2}&x_{2p}x_{2q}\\ \vdots&\vdots&\vdots&\vdots&\vdots&\vdots\\ 1&x_{Mp}&x_{Mp}^{2}&x_{Mq}^{2}&x_{Mp}x_{Mq}\end{array}\right] \tag{11}\]  
The method of least squares analysis of multiple-regression solves the equations as follows [21; 29; 30]:  
\[a=\left(A^{T}A\right)^{-1}A^{T}Y \tag{12}\]  
This equation creates all the coefficients of Equation (10) for all the M triple sets [21; 29; 30].