## 3 Surrogate Modeling
### Surrogate Model  
A surrogate model of inter-ply shear stress was developed. The developed model has a functional dependence on the aforementioned forming parameters. To approximate multidimensional functions, surrogate models such as the radial basis function are often used. The commonly known radial basis functions are given in Table 3.  
In this study, multi-quadratic type radial basis functions (MQ-RBFs) are found to well approximate the functional form of inter-ply shear stress. Therefore, the MQ-RBFs are for the construction of the surrogate model. \begin{table}
\begin{tabular}{c c} \hline \hline
**Radial Basis Function** & **Mathematical Expression** \\ \hline Multi-quadratic & \(\sqrt{1+c^{2}\sum_{k=1}^{n}(x^{k}-x_{i}^{k})^{2}}\) \\ Inverse multi-quadratic & \(\frac{1}{\sqrt{1+c^{2}\sum_{k=1}^{n}(x^{k}-x_{i}^{k})^{2}}}\) \\ Gaussian & \(e^{-c^{2}\sum_{k=1}^{n}(x^{k}-x_{i}^{k})^{2}}\) \\ Inverse quadratic & \(\frac{1}{1+c^{2}\sum_{k=1}^{n}(x^{k}-x_{i}^{k})^{2}}\) \\ Thin plate spline & \((x^{k}-x_{i}^{k})^{2}ln(x^{k}-x_{i}^{k})\quad m=2,4,6,...\) \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Commonly known radial basis functions. Radial Basis Function  
The exponential convergence, dimension insensitivity, mesh, and geometry-free features of RBFs draw much attention in various research disciplines. These bases functions can be categorized into two: finitely and infinitely differentiable functions. Finitely differentiable RBFs depend only on Euclidean distance between two grid points, whereas infinitely differentiable RBFs depend on both the shape parameter and Euclidean distance between two grid points. For the infinitely differentiable type of functions, shape parameter selection remains unsolved. An unrealistic value of this shape parameter affects the accuracy and stability of its RBF by either flattening (invoking the ill-possedness of the matrix) or sharpening (underestimating the effective arrangement of the control point). Radial basis functions have the following form,  
\[f(x)=\sum_{i=1}^{N}\lambda_{i}\phi(\mid x-x_{i}\mid c) \tag{1}\]  
where \(\phi(\mid x-x_{i}\mid c)\) are user defined basis functions. The variables \(x\), \(x_{i}\), and \(c\) imply interpolation points, control (center) points, and the shape parameter, respectively. The form of this basis function can be of any type as long as its positive definiteness is confirmed. Based on the type of RBF used, the shape parameter may or may not be required. \(\lambda_{i}\) are weight parameters that can be obtained from the linearly independent system of equations. The implemented MQ-RBFs have the form:  
\[\phi(\mid x-x_{i}\mid\mid c)=\sqrt{1+c^{2}\sum_{k=1}^{4}(x^{k}-x_{i}^{k})^{2}} \tag{2}\]  
where \(k\) stands for the number of independent design parameters and \(i\) represents the known data points, respectively. The approximating function and the inter-ply shear stress should have the same numerical values at least at the known data points. Based on this restriction, there are 45 independent equations where each equation is a linear combination of the non-linear MQ-RBFs. These equations can be written in matrix form as follows:  
\[ \begin{bmatrix}\phi(\mid x_{1}-x_{1}\mid\mid c)&\phi(\mid x_{1}-x_{2}\mid\mid c )&.&.&.&\phi(\mid\mid x_{1}-x_{N}\mid\mid c)\\ \phi(\mid\mid x_{2}-x_{1}\mid\mid c)&\phi(\mid\mid x_{2}-x_{2}\mid\mid c)&.&.&.& \phi(\mid\mid x_{2}-x_{N}\mid\mid c)\\.&.&.&.&.&.\\.&.&.&.&.&.\\.&.&.&.&.&.\\ \phi(\mid\mid x_{N}-x_{1}\mid\mid c)&\phi(\mid\mid x_{N}-x_{2}\mid\mid c)&.&.&.& \phi(\mid\mid x_{N}-x_{N}\mid\mid c)\end{bmatrix}\begin{bmatrix}\lambda_{1} \\ \lambda_{2}\\ \lambda_{2}\\.\\.\\ \lambda_{N}\end{bmatrix} = \begin{bmatrix}f_{1}\\ f_{2}\\.\\.\\.\\ \lambda_{N}\end{bmatrix}  \tag{3}\]  
The compact form of the above matrix is given as:  
\[[\Phi][\Lambda]=[F] \tag{4}\]  
where \([\Phi]\) is a square matrix, \([\Lambda]\) is the column vector of the weight parameter, and [F] is the column vector of the known output values of inter-ply shear stress, respectively.