Level-set-based topology optimization of a morphing flap as a compliant mechanism considering finite deformation analysis

Abstract

A morphing flap, a morphing wing section, can deform its geometrical shape seamlessly and continuously to improve the aerodynamic performance. To obtain a compliant morphing flap internal mechanism to realize the target deformation, this study aimed to develop a nonlinear topology optimization method based on a level-set method. However, numerical instability occurs during the design optimization process within the scope of finite deformation analyses. Accordingly, the numerical instability problem caused by an artificial weak material representing a void region in the optimization algorithm was resolved. In particular, this problem was resolved by structural modeling without such weak materials using the mesh adaptation method and by installing a spring component with a negligibly small spring constant. Subsequently, a finite deformation analysis was performed and integrated with the level-set-based topology optimization model. Finally, the proposed method was validated by solving the morphing flap design problem through numerical analyses, wherein an optimum structural configuration under finite deformation was obtained.

Appendix A: Discussion of influences of mesh quality on optimal design result

In this section, we investigated the influence of various qualities of mesh depending on the specific structural topology. While meshes generated under identical configuration conditions by the mesh adaptation method with the Mmg open-source library (Dapogny et al. 2014) were reproducible, the mesh quality might vary depending on the structural form. To investigate this, the finer mesh was generated for multiple structural forms in the optimization process by changing the setup conditions in Mmg, and the analysis results were compared. Then, its influence on the optimal design results was discussed. We prepared the finer mesh obtained for the same level set function distribution (i.e., the same structural topology) by the mesh adaptation method described in section 4.1 with different setting conditions $h_{min}$, as shown in Fig. 18. The mesh quality generated by Mmg depends mainly on the four user-defined parameters: $\varepsilon _{mmg}$, which is the tolerance over the geometric approximation of structural boundary $\partial \Omega$; $h_{min}$ and $h_{max}$, which are the minimal and maximal authorized size for an edge of the generated mesh $\widetilde{{\mathcal {T}}}$, respectively; $h_{grad}$, which is a control parameter for the variation of edge lengths among $\widetilde{{\mathcal {T}}}$. Here, a lower minimum length of element edges produces a higher quality mesh. In the following examples, we assume that the smaller the change in the analysis results when different meshes are used, the more accurate the original mesh was.

Fig. 18

Comparison of the original and finer meshes with $n_e$ number of elements obtained by the mesh adaptation method under different conditions for the 125th structural topology during optimization: (a) $(h_{min},\;h_{max})=(1.0,\;2.0)$ mm (b) $(h_{min},\;h_{max})=(0.5,\;2.0)$ mm

For the $50\textrm{th}$, $75^\textrm{th}$, $125^\textrm{th}$, $150^\textrm{th}$, $200^\textrm{th}$, and $300^\textrm{th}$ structural topologies in Fig. 12, a finer mesh was generated using the same settings as in Fig. A. The values of the objective function f were then compared in Fig. 19 between those meshes and the original mesh. For all structural topologies, differences in mesh quality were found to be within a few percent of the relative error of the objective function f. Then, the influence of different mesh quality on updating the level set function $\phi ({\varvec{X}})$ was examined for the $150^\textrm{th}$ and $300^\textrm{th}$ structural topologies, which have relatively large relative errors. The level set functions were updated based on the analysis results using two different quality meshes, and they are compared in Fig. 20. There, the level set function distribution before the update and the difference in distribution between the two updated results are shown. In the left figure, red and blue indicate the structure and the void, respectively, while in the right, blue and red indicate the locations where the mesh difference had a significant influence on the level set function update. The influence of mesh quality was particularly large on substructures where deformation was expected to occur, such as near the wing skin and on the strangled substructure.

Fig. 19

Relative error of the objective function value f using a finer mesh, where the value calculated using the original mesh was used as the basis

Fig. 20

Influence of mesh quality on level set function update: (left) Distribution of level set functions before updating; (right) Difference between level set function update results $\phi ({\varvec{X}})^{F}$ and $\phi ({\varvec{X}})^{O}$ based on analysis results using finer and original meshes ($\phi ({\varvec{X}})^{F}-\phi ({\varvec{X}})^{O}$)

Finally, the influence of mesh quality on the overall optimization was investigated. For the compliant morphing flap design problem, topology optimization with different conditions using the mesh adaptation method was conducted. Then, obtained optimal solutions were compared to each other. The optimal solutions were shown for three conditions as $(h_{min},\;h_{max})=(1.0,\;2.5),\;(1.0,\;2.0)$, and $(0.5,\;2.0)$ mm. The structural configurations and the deformed shapes are shown in Figs. 21 and 22. The resulting structural configurations had similar feature structures, such as a connection with the underside of the wing skin to transmit the input load and thickness of the wing skin and cavity in the upper surface to passively adjust the bending deformation. However, due to differences in mesh quality, their structural configurations were a little different from each other, especially in the elongated substructures (Table 3).

Fig. 21

Comparison of optimal structural configurations for three different conditions in the mesh adaptation method

Fig. 22

Comparison of the deformed shapes obtained by analysis (Case 1)

Table 3 Comparison of objective function $f_i$ ($i=1,\;2,\;3$) for three different conditions in the mesh adaptation method