# Gradient-Based Aerodynamic Optimization of an Airfoil with Morphing Leading and Trailing Edges
## 2 Methodologies
### Feasible Morphing Airfoil Geometry Parameterization
#### 2.2.3 Morphing Leading and Trailing Edge Algorithm  
Many of the the morphing concepts applied to transport aircraft are now trying to improve existing aircraft by retrofitting the leading edge [20], the trailing edge [21; 22], or the upper surface [23], without changing the main bearing structure [24]. This can be explained from both an aerodynamic and a structural point of view. On the one hand, the aerodynamic performance is more sensitive to the geometric variation at the leading and trailing edges. The stagnation point, where the local velocity turns to zero and the static pressure shows the highest value, is located at the leading edge. Additionally, when the airfoil approaches the stall angle, the separation of flow starts on the upper surface of the leading edge. Besides, the laminar-to-turbulent flow transition point and the shock wave are located on the upper surface [25]. On the other hand, the wing-box of a fix-wing aircraft is the primary load-carrying structure. Designers intend to introduce the morphing system while keeping the overall static and dynamic performance of structure [26]. The distinction between standard aerodynamic shape optimization and morphing airfoil optimization is obvious. In the morphing airfoil optimization, the morphing structure designed to form morphing shape should be considered during the shape optimization. In the preliminary morphing optimization framework, the feasible morphing should be defined based on certain morphing mechanisms, or at least the behavior of the skin that is part of the morphing mechanisms. The skin length \(L(\psi)\) is also useful in representing the kinematics of a morphing skin. For the passive non-stretched skin, the skin length \(L(\psi)\)should maintain a constant value. For those actuated surfaces, morphing skin coupled with a sliding system and the advanced flexible morphing skin [27], the variation of skin length \(\Delta L(\psi)\) can be released to a particular value [8]. This paper adopts a feasible morphing airfoil parameterization method aimed at providing local deformation, skin length control, and in the meantime maintaining invariance of wing-box region. The feasible morphing airfoil generation algorithm based on the CST parameterization method is shown in Figure 3, including an inverse fitting and a single-value optimization problem. The former guarantees the generated airfoil satisfying the wing-box constraint, and the latter provides length control of the skin. Here, the parameters in the CST airfoil parameterization method are divided into three categories based on the aerodynamic performance index, skin length constraint, and wing-box constraint. For example, consider an airfoil shown in Figure 2. The chord-wise position \(x_{\text{LE}}\) plays a key role in keeping the constant skin length when leading edge is morphing. In addition, design variables driven by the aerodynamic optimizer are the leading edge radii of upper and lower skin (\(R_{\text{LE},\text{u}}\), \(R_{\text{LE},\text{i}}\)), vertical position (\(z_{\text{LE}}\)), and a zero or one CST coefficient that peaks near the front spar (\(A_{u,0}\) and \(A_{l,0}\)). Regarding the trailing edge morphing, the chord length is determined in order to provide skin length control. The parameters exposed to the aerodynamic optimization problem are the boat-tail angle \(\beta\), vertical position \(z_{\text{TE}}\) of trailing edge, and several CST coefficients with peaks located in the trailing edge (\(A_{u,i}\) and \(A_{l,i}\) for \(i=7,8\)). In the nested single value optimization problem, a convergence tolerance of \(1\times 10^{-6}c\) is adopted. The remaining CST coefficients are obtained by solving the inverse fitting problem Equation (7) so that the curves always attach the wing-box. Let \(\psi_{\text{b}}=\left\{\psi_{l},\,\zeta_{l}\right\}_{j\in\Omega_{b}}\) be the points in the subset space of airfoil \(\Omega\), where subscript \((\cdot)_{\text{b}}\) stands for box. The morphing part of airfoil \(\Omega\) is indicated by subscript \((\cdot)_{\text{m}}\).