## 2 Method
### Objective Function Definition  
Numerous researchers have stated that there is always a trade-off between aerodynamic coefficients and performance parameters in aerodynamic design problems [66]. A difficult balance exists between increasing aircraft performance and meeting design constraints efficiently in practical aerodynamic design problems. For example, the objective of aerodynamic optimization is lift maximization, drag minimization or maximization of the lift-to-drag ratio while keeping \(C_{L}\) and/or \(C_{D}\) constant as constraints. Likewise, various airfoil shapes are optimized for their morphing by considering an increased lift-to-drag ratio and, at the same time, for meeting the given constraints. Other airfoil shape optimizations may involve those of multiple aerodynamic coefficients based on performance parameters such as range, loiter and control. Such optimization techniques can be applied to solve complex aerodynamic design problems for a broad range of flight conditions. In addition, the design problem shown in this paper must consider the extended range of flight operating conditions that a UAV is expected to encounter within its flight envelope. As mentioned above, this study considers the optimization of the airfoil shape for drag minimization and for endurance maximization as individual target objective functions. The mathematical formulation of the drag minimization objective function is the following. \[minimize\ C_{D}(x);x\ e\ (airfoil\ set) \tag{1}\]  
Subject to  
\[C_{L_{max}}\ (x,\ a,\ M\ \geq)1\cdot 608C_{L_{maxpl}}\geq C_{L_{min, baseline}}(x,\ a,\ M)\]  
Figure 2: Schematics of the optimization procedure. where  
\[x\in(airfoil\ set,the\ vector\ of\ airfoil\ shape\ function\ design\ variables)\]  
The lift coefficients of the UAS-S45 are calculated for each angle of attack using design parameters mentioned in a previous paper of our team [67]. The constraints implemented are both maximum lift coefficient and lift coefficient of baseline airfoil at each angle of attack. The optimization runs are executed with a minimum lift coefficient at each angle of attack. Constraints are applied by penalty functions to the objective function to ensure optimum performance is obtained at each given angle of attack. For the design of a long-endurance UAV, Equations (2) and (3), which are needed to determine the endurance (\(E\)) of a propeller-powered UAV, are well established using the Breguet formula [68]. \[E=\frac{\eta_{pr}}{c_{p}}\frac{c_{L}^{3}}{C_{D}}\sqrt{2\rho S}\left(\frac{1}{ \sqrt{W_{f}}}-\frac{1}{\sqrt{W_{i}}}\right) \tag{2}\]  
where \(c_{p}\) is the specific fuel consumption, \(\eta_{pr}\) is the propeller efficiency, \(\rho\) is the free stream density, \(S\) is the wing planform area, \(W_{i}\) is the initial weight of the aircraft, \(W_{f}\) is the final weight of the aircraft and \(C_{L}\) and \(C_{D}\) are the lift and drag coefficients, respectively. The parameters that do not affect the airfoil shape design in the endurance performance can be neglected in Equation (2), and the parameters that can affect the airfoil shape design are only the two aerodynamic coefficients \(C_{L}\) and \(C_{D}\); therefore, Equation (2) can be reformulated as  
\[E_{a}=\frac{C_{L}^{\frac{3}{2}}}{C_{D}} \tag{3}\]  
Therefore, an optimized airfoil maximizes the lift and minimizes the drag coefficients in the above Equation (3); therefore, the overall endurance aerodynamic efficiency is maximized. The weighted average of the flight envelope is used to find out the time that the UAV would have to spend in different flight regimes. Since the present study concerns the surveillance UAS-S45 optimization, it is assumed that 80% of total flight time is spent in loiter flight conditions, which require maximum endurance.