## 4 Optimization Analysis
Of course, if a sample appears many times during the evolution process, it one of the best samples. Therefore, by relating the number of samples (w.r.t to total number) to a specific range, it is possible to quickly and easily understand which regions should be chosen for the hinges. The graphs show that the most feasible values of the H1-X variable are concentrated in the range of 100-450 mm. For the variable H2-X, there is a thickening in the region 800-1080 mm. The results indicate that increasing the distance between the hinges is a valid option to reduce the bearing forces in the four points both for deployed and folded configurations. The H1-Z and H2-Z variables are quite obvious and the best value, for both of them, results in the smallest feasible value, that is 10 mm. Figure 14: Objective function time history in the folded configuration. (**A**) forward hinge in upper location; (**B**) forward hinge in lower location; (**C**) afterward hinge in upper location; (**D**) afterward hinge at a lower location. Figure 15: Probability Density Function of feasible designs. Simultaneous representation of all eight objective functions is not possible, therefore the last one has been divided into two groups. Each group reports the objective functions of its wing configuration. The 4D bubble graph of Figure 16 shows the objective functions relating to the deployed wing (only feasible designs). The horizontal and vertical axes report the bearing loads of the upper pin for forward and afterward hinge respectively, the bubble color refers to the bearing load of the lower pin of the forward hinge and, finally, the bubble size is related to the bearing load of the lower pin of the afterward hinge. The figure also highlights the region (which is part of the Pareto front) where individuals, even if they do not minimize a specific objective function, are a good compromise in the minimization of the four bearing loads (all bearing loads are about 20 kN). The bubble-graph 4D of Figure 17 shows the objective functions relating to the folded wing (only feasible designs). In this case, it is evident that the selection of the optimal individuals is simplified since many individuals are able to simultaneously minimize the four objective functions. Many of the individuals belonging to the region highlighted in Figure 16 (deployed configuration) are also highlighted in Figure 17 (folded configuration). Figure 18 shows only the elements present in both regions. About 13 designs can be considered as the best ones for both configurations. Figure 16: Pareto front of the deployed configuration. Figure 17: Pareto front of the folded configuration. Figure 19 shows the individuals present in both regions defined above, and, in particular, it is possible to relate any selected individual to the design variables. This graph highlights again that the objective functions are minimized as the relative distance between the two hinges increases. All selected designs provide bearing loads close to each other and therefore any of them is not clearly better than another one. Therefore, the final choice could be completed considering other functional requirements, such as the real possibility of installing the hinges in the selected regions. The obtained results enable a preliminary estimation of the diameters of the bolts/pins and the plate thickness. For the subsequent evaluations, only the bearing components were taken into account as they are strictly connected to the sizing of both the locking pins (in the unfolded and folded configuration) and the rotation axes. Because of the mission of the investigated vehicle and, therefore, due to the operative temperature, the pins, the lag, and the hinge axes are made of titanium alloy TiGAl4V. Considering an operative temperature equal to 160 \({}^{\circ}\)C, the tensile ultimate strength is equal to about 720 MPa, the shear ultimate stress is about 398 MPa and the bearing stress is 1071 MPa.