# Design of a Variable-Stiffness Compliant Skin for a Morphing Leading Edge
## 4 Optimization Method  
### FEM Model  
To decrease material strain level, a pure bending principle is used here, which means there is no compression or stretching in the middle plane of the compliant skin, which still suffers severe bending strain. Therefore, this study employs an aviation-level woven glass fiber reinforced polymer (SW100A/6511, Weihai Guangwei Composites Co., Ltd., Weihai, China), which is characterized by high ultimate strain. The mechanical properties of this material are presented in Table 1.  
Considering the complexity of the manufacturing process, the variable stiffness is realized by changing layup thickness and ply orientation of straight fiber layups in different regions. Taking the manufacturing realizability and deformation accuracy into account, the compliant skin is partitioned as 10 uniformly spaced thickness regions along the circumferential direction as shown in Figure 6. In each FEM analysis called for by the main optimization procedure, the internal mechanism can be simulated with actuated forces at the four respective interface points during optimization. As the skin deformation depends on the layup and the magnitude of the actuated forces, the layup and the forces all need to be optimized and defined as design variables. However, the variation range of the actuated forces is not certain in advance. Therefore, a series of trial calculations are needed to obtain the approximate ranges of the actuated forces, which are utilized as the optimization constraints and presented as Equation (5) in Section 4.2. Once the design variables of the actuated forces are given in every optimization step, the force boundary in every single analysis is certain. It is worth mentioning that a nonlinear finite element analysis, rather than a linear one, is employed to help produce a movement locus of the interface point above each stringer hat during the morphing process, which strongly supports later inner kinematic design. \begin{table}
\begin{tabular}{c c} \hline \hline
**Property Parameter** & **Value** \\ \hline Elastic modulus along fiber direction (GPa) & 23.20 \\ Elastic modulus transverse to fiber direction (GPa) & 23.20 \\ Poisson’s ratio & 0.12 \\ Shear modulus (GPa) & 2.97 \\ Tensile strength along fiber direction (MPa) & 477.00 \\ Compressive strength along fiber direction (MPa) & 302.00 \\ Tensile strength transverse to fiber orientation (MPa) & 477.00 \\ Compressive strength transverse to fiber orientation (MPa) & 302.00 \\ Shear strength (MPa) & 54.00 \\ Tensile ultimate strain transverse to fiber orientation (\(\mu\)) & 33,166 \\ Compressive ultimate strain transverse to fiber orientation (\(\mu\)) & 13,538 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Mechanical properties of the woven glass fiber reinforced polymer laminate. Figure 5: Structural concept of the morphing leading edge. Figure 6 presents the finite element model of the variable-stiffness compliant skin. In this study, the nonlinear FEM solver in Abaqus of Dassault Systems is used and called by a python script, which is an independent module for the main optimization procedure. For the nonlinear FEM solver, an automatic type of step increment is adopted with the maximum number of increments of 100. The initial, minimum, and maximum increment sizes are set as 1, 1 \(\times\) 10\({}^{-5}\), and 1, respectively. The full Newton method is used as the solution technique for the FEM model. Finally, the solver gives the coordinates of every single node on the complaint skin, which are used to calculate the objective function of the compliant skin optimization problem. The nonlinear finite element analysis illustrated in this study is referred to as a geometrically nonlinear analysis. Therefore, the magnitude of the enforced forces is constant, while their direction is always changing, which means that only the direction of the distributed pressure should always be updated.