# Gradient-Based Aerodynamic Optimization of an Airfoil with Morphing Leading and Trailing Edges
## 3 Results and Discussion
### Sensitivities Verification  
The gradients obtained with the method provided in this paper are compared with those computed with forward-difference. Different finite-difference steps have been tested, while decreasing from a step size of \(1.0\times 10^{-3}\) of chord length until the finite-difference gradients converged. In the results shown in this subsection, the reference gradients were computed using a finite-difference step size of \(1.0\times 10^{-7}\). Surface sensitivities, which are the variations of the objective function with respect to the infinitesimal deformation along the normal direction of surface nodes, are projected to the feasible morphing airfoil geometry parameterization design space through the geometric sensitivities matrix to obtain gradients. Before demonstrating the gradients, it is worthwhile showing the geometric sensitivities projection matrix. Geometric sensitivities, also known as design velocities, are computed through forward finite differences. A step of \(1.0\times 10^{-5}\) was used. The preliminary test indicates that the geometric sensitivities are insensitive to the size of the finite difference step. Figure 6 provides the geometric sensitivities for an airfoil of which wing-box locating \(\psi_{\text{b}}\in[0.1,0.7]\) with respect to \(R_{\text{u}}\), \(R_{\text{l}}\) and \(z_{\text{LE}}\) for leading edge morphing, \(z_{\text{TE}}\), \(\beta\) and two extra CST coefficients \(A_{\text{8}}\) and \(A_{\text{9}}\) on upper and lower surface for the trailing edge morphing. Note that arrows with outward displacement on the airfoil indicate positive geometric sensitivities, negative for inward. A scale factor has been applied to the magnitude of each geometric sensitivity for a better illustration. Theoretically, the morphing leading and trailing edge algorithm ensures the perturbation appears only in the design region. However, a small error is observed in the non-morphing domain. This error is due to the residual that is introduced in solving the over-determined equation. The magnitude of residual is \(1.0\times 10^{-5}\) times smaller than the required value of geometric sensitivity, which is considered as acceptable. The geometric sensitivities at the morphing leading edge region shown in Figure 6a-c have the same order of magnitude. On the contrary, the magnitudes of geometric sensitivities for the morphing trailing edge, from Figure 6d-i, show inconformity. The impact of \(z_{\text{TE}}\) is 200 to 300 times larger than other parameters. This result suggests that gradients are highly sensitive to the vertical displacement of the trailing edge, which may amplify the error of adjoint results. The sensitivities on the surface nodes provided by continuous adjoint and gradients comparison are displayed in Figure 7. It is not surprising that a discontinuity of sensitivity at the tip of the trailing edge on the upper surface is present, as shown in Figure 7a, which leads to the gradient inaccuracy at the gradient concerning the design variable for trailing edge morphing. Several authors [34; 37] also observed this phenomenon, mainly due to the assumption of smooth surface in continuous adjoint method and the presence of the sharp trailing in this case. Readers who are interested this problem are prompted to refer to [33; 38] for details. Fortunately, except for discrepancies for design variables at the trailing edge, no significant problem has been found. The gradients with respect to the design variables of leading edge show good agreement with the results from finite differences. Although the numerically exact gradient information was not obtained, the gradients still can provide sufficient accuracy and lead the optimizer to find optimized results. Figure 6: Geometric sensitivities of morphing leading and trailing edges. Geometric sensitivities of morphing leading edge with respect to \(\text{R}_{\text{u}}\), \(\text{R}_{\text{l}}\), and \(z_{\text{LE}}\), whose amplitudes have been multiplied by a factor of 20 for better demonstration.