# Optimization of DLTS Hinges for the Assembly of the Solar Arrays of a Communication CubeSat
## 4 Results and Discussion
### Optimization Analysis--RSM Model  
The optimization analysis was set for the maximization of the steady-state moment, where the maximum \(\sigma_{\rm v}\) was applied as a constraint. The excessive \(\sigma_{\rm v}\) could generate permanent damage on the DLTS hinge and prevent deployment performance. Therefore, the constrained max \(\sigma_{\rm v}\) was maintained below the allowable \(\sigma_{\rm v}\) of the DLTS hinge material. The steady-state moment was established to be an optimized objective as it is a major characteristic for the maneuverability of the hinge. The \(\theta\) and \(t\) variables are identified as critical deign parameters through the primary and secondary sequence study. The \(R\) parameter was negligible as it did not directly impact the objective and was maintained at 2 mm. The optimized decisions are illustrated by the following mathematical configurations and symbols [25]. \[\text{Objective}=Ms\ (\text{maximization})\]  
\[\text{Constraint}:\ \ \sigma_{max}\leq\overline{\sigma}\]  
\[\text{Critical parameters}:\ \underline{x_{i}}\leq x_{i}\leq\overline{x_{i}}\]  
\[i=1,\ldots,\ n \tag{2}\]  
where,  
\(Ms\) = steady-state moment of DLTS hinge  
\(\sigma_{max}\) = maximum \(\sigma_{\rm v}\) in stowed state  
\(\underline{x_{i}}\) = lower limit of the critical design parameter  
\(\overline{x_{i}}\) = upper limit of the critical design parameter  
\(n\) = number of total critical design parameters  
Response surface methodology/RSM was selected for this engineering optimization application. RSM forms an approximate representation where a relation is acquired between the critical design parameters and objectives by employing first and second-order polynomial functions [17]. The RSM approach was carried out by varying the \(t\) and \(\theta\) parameters from 0.06 mm to 0.24 mm, and 54\({}^{\circ}\) to 94\({}^{\circ}\), respectively. The sample points were extracted from the two four-factor methods. The \(\sigma_{\rm v}\) and steady-state outcomes were explored by utilizing computational simulation results. The following polynomial equation was employed for the acquisition of the undetermined parameters and, consequently, the obtainment of the function values of the \(\sigma_{\rm v}\) and steady-state moment [24]. \[F(x)=a_{1}+\sum_{i=1}^{n}a_{i}+1x_{i}+\sum_{i=1}^{n-1}\sum_{j=i+1}^{n}a_{ij}x _{i}x_{j}+\sum_{i=1}^{n}a_{ii}x_{i}^{2} \tag{3}\]  
where,  
\(F(x)\) = estimation of the function's true response  
\(a\) = undetermined parameters  
The third order of the polynomial equation was utilized and was run through MATLAB R2021b version to obtain the objective and constraint function, which showcased the yielding values of the steady-state moment and \(\sigma_{\rm v}\). The absolute value of the objective and constraint function was determined to be 0.00109 and 0.00008, respectively. The relative error value between MATLAB and the simulation for the objective and constraint function was estimated to be 8.23% and 5.67%, correspondingly. The accuracy of the objective and constraint function values were assessed by implementing the complex (\(R^{2}\)) and modified complex (\(R_{adj}^{2}\)) correlation coefficients, which are preferable when close to one another for precision and accuracy of the function [18]. \(R^{2}\) and \(R_{adj}^{2}\) were proven by the following deviations [26]. \[R^{2}=\frac{1-\sum_{i=1}^{m}(y_{i}-\tilde{y}_{i})^{2}}{\sum_{i=1}^{m}\left(y_{ i}-\widetilde{y}\right)^{2}} \tag{4}\]  
\[R_{adj}^{2}=1-\frac{m-1}{m-N}\left(1-R^{2}\right) \tag{5}\]  
where,  
\(\widetilde{y}_{i}\)= response of sample point with ith terms (FEA)  
\(\widetilde{y}\)= mean response  
\(\tilde{y}_{i}\)= response of sample point with ith terms (MATLAB)  
\(m\)= number of design points  
\(N\)= number of basic functions  
\(R^{2}\) and \(R_{adj}^{2}\) for the max \(\sigma_{\mathrm{v}}\) response were obtained to be 0.99453 and 0.99324, with a percentage difference of 0.12%.