## 3 Numerical Examples
### Engineering Example--Morphing Wing Design  
The numerical results culminate with a practical example for the application of the introduced methods. In this case, the internal structure (compliant mechanism) for the deformation of the morphing leading edge of a sailplane wing is designed. The airfoil and wing planform for the morphing wing concept is designed by [54; 55] and its elastic wing shell is described by [56]. As sailplanes have a large operation envelope, the ability to morph the wing results in drastic performance gains. Figure 10 shows a schematic problem definition for a compliant mechanism rib to be used in a droop-nose morphing wing concept. A compliant mechanism that displaces an output node on the outer leading edge contour in a predefined position \(u_{\text{out}}\) (problem type II) is sought. Combining multiple  
Figure 7: Path generating example mechanism topology with smoothed boundary based on Figure 6f: \(u_{\text{out,x}}=9.09\) mm, \(u_{\text{out,y}}=-1.65\) mm, \(\max(\bar{\sigma})=53.7\) MPa (5283 solid elements/0 void elements). mechanisms of this type along the leading edge contour allows the wing to be morphed accurately to a target airfoil. The mechanisms defining an airfoil are stacked together in span direction and connected by a belt along the leading edge contour. By this means, a single rib is formed, which can be bonded to the wings shell. In span direction of the wing multiple of these stacked ribs are used, considering the changing airfoils along the wing. The optimization problem defined in Equation (65) is used to synthesize such mechanisms. Figure 8: Path generating example solutions with and without multiresolution topology optimization. Figure 9: Output paths for path generation mechanism example solutions in Figures 6 and 7. In Figure 10, the gray area represents the designable domain, whereas black and white areas represents non-designable solid or void, respectively. To the upper right, fixed supports represent an adhesive fixation of the mechanism to the wings spar. The mechanism is activated by the application of a positive x-displacement \(u_{\text{in}}\) at the input node, which is guided by a support in the y-direction (shown in yellow). The design space is discretized by bilinear quadrilateral finite elements with four Gauss integration points. Material and optimization parameters are used according to the path generation example shown in advance. Only one precision point prescribes the target position of the output node in the activated, i.e., morphed state (see Table 6). The counter load cases are set according to Table 7 simulating aerodynamic loads on the mechanism acting against the direction of deformation. In Figure 11, the deformed results for the compliant mechanism rib are compared for the various optimization methods used. In Figure 11a, the problem was solved without consideration of stress constraints. It can be observed that a low objective function value is reached as this design provides a good approximation of the target displacement. On the other hand, a practical application of the mechanism is excluded due to the high stresses in the flexible hinges connecting the mechanism to the support. These stresses would lead to failure of the mechanism. In Figure 11b, stress constraints were also considered. The stresses are reduced to the admissible level at the cost of a slightly higher deviation from the exact target displacement. A mechanism suitable for practical application could be obtained. Figure 11c,d show respectively the results for the volume-constrained as well as the volume- and stress-constrained rib mechanism obtained by the MTOP method. Therefore, the same FE mesh is used as in Figure 11a,b but with a four times higher resolution of the density mesh. By this means, similar topologies are obtained as for the standard topology optimization in Figure 11a,b but at a higher resolution. The obtained results are hence