# Design of a Variable-Stiffness Compliant Skin for a Morphing Leading Edge
## 4 Optimization Method
### Optimization Model  
As illustrated before, the design objective for the morphing leading edge is to obtain the target profile as precisely as possible (Figure 7). To describe the deviation between the realized final profile and the target profile, unlike what was used in previous work [21,22], an improved objective function based on weighted least square error (WLSE), shown in Equation (1), is proposed for more delicate deviation control. The previous least square error (LSE) may be insufficient since the final LSE can be low enough even when unacceptable deformation deviation in a certain aerodynamically critical regime, such as the frontmost tip, still exists. Therefore, a weighted penalty is introduced to the LSE formula. \[WLSE=\sum_{i=1}^{n}w_{i}\frac{d_{i}}{n}=\sum_{i=1}^{n}w_{i}\frac{\sqrt{\left(x _{i}-x_{i}^{*}\right)^{2}+\left(y_{i}-y_{i}^{*}\right)^{2}}}{n} \tag{1}\]  
In the above, \(n\) is the number of monitoring points; \(w_{i}\), \(d_{i}\), \(\left(x_{i},y_{i}\right)\), and \(\left(x_{i}^{*},y_{i}^{*}\right)\) are the weight, the deviating distance from the target, the actual coordinate, and the target coordinate of the \(i\)th monitoring point, respectively. The weight coefficient follows a bi-linear law that is empirically generalized in Equation (2) over \(n\) = 20 uniformly spaced monitoring points. In Equation (2), s is the normalized curve length along the circumferential direction of the profile. \[\left\{ \begin{array}{l}w=0.3+0.66667s\ for\ 0<s\leq 0.6\\ w=1.3-s\ for\ 0.6<s\leq 1\end{array} \right. \tag{2}\]  
Figure 6: Finite element model of the variable-stiffness compliant skin. The design variables include layup sequence in each design zone, each stringer's position, and actuating force at each interface point. To simplify the optimization process, all design zones are assigned with the same layup pattern [0\({}_{\text{a}}\)/45\({}_{\text{b}}\)/90\({}_{\text{c}}\)/-45\({}_{\text{d}}\)/90\({}_{\text{e}}\)]s, while virtual thicknesses a, b, c, d, and e vary from one zone to another (0\({}^{\circ}\) degree aligns with the circumferential direction of the airfoil and 90\({}^{\circ}\) aligns with the spanwise direction). The constraint of variables a to e is that each of them must be an integral multiple of a single sheet. Obviously, this layup sequence is not fit for manufacturing, and a layup adjustment process is essential (discussed in Section 3). Incidentally, the layup sequence of four stringers is set uniformly as [0/45/90/-45/90]s. Consequently, there are total 62 design variables: five variables defining layup sequence in each of the 10 design zones, the position of four stringers in terms of normalized length, and actuating force components in x and y directions in each of the four actuating points. Furthermore, based on the pure bending principle, the strain in the skin results from pure bending, and the relationship between the maximum strain at the outmost skin surface and the curvature and thickness of the skin at that point is given in Equation (3):  
\[t(s)=\frac{\varepsilon_{\text{lim}}}{1/2\Delta\kappa(s)} \tag{3}\]  
where \(\varepsilon_{\text{lim}}\) is the material limit in terms of strain, \(s\) is defined as the normalized length along the skin profile from the lower end to the upper, \(t\) is the maximum allowable thickness, and \(\Delta\kappa\) is the curvature variation. Both \(t\) and \(\Delta\kappa\) are a function of \(s\). According to the magnitude of the curvature variation between the clean case and drooped case, the local maximum skin thickness is determined from Equation (3) and then used as one of the constraints in the following skin stiffness distribution optimization process. Consequently, as for the selected GFRP, the maximum skin thickness around the circumferential length of 42% and 50%, where the greatest curvature variation occurs, must be less than 3.6 mm. This is used as one of the optimization constraints.