Next, we will test the tow-steered wingbox optimization for the presence of local minima. While gradient-based optimization techniques guarantee rapid convergence to a minima of the optimization problem, there is no guarantee that the converged minima is in fact a global minimum. This problem of local optimality is a well-known weaknesses of gradient- based optimization methods and can lead to difficulties in improving design performance if the problem is severely multi-modal. Multi-modality in aerodynamic shape optimization for wing designs has already been explored in great detail by Lyu et al. [79] and Bons et al. [110]. Both of these studies concluded that when the necessary physics are modeled and appropriate design variables and constraints are applied, the design problem can be reduced to a single-modal optimization problem featuring only one global minimum. This gives us confidence that that the aerodynamic discipline will not add multi-modality to the aerostructural design optimization problems presented in the following sections. The goal of this study is to perform a local minimum study for a tow-steered wing design optimization on the structural discipline side as well. Of primary concern are the tow angle variables θcp0 , because of the inherent multi-modal nature of the trigonometric functions (sin, cos, etc.) used to rotate the stiffness matrices for each ply in the classical laminate theory (CLT) formulation. If through structural optimization we can verify that multi- modality is not a concern for the wing design considered throughout the remainder of this thesis, then we can extend this conclusion to remaining aerostructural optimizations as well.
To check for local minima and multi-modality we will begin the optimization from four different starting points: the initially unsteered layup 0◦ layup that was the starting point for the the steered optimization presented in the previous section and three randomized ini- tial tow-steered patterns. For the random starting points the design variables corresponding to the tow-path control points, θcp0 , are set to random values at the beginning of the opti- mization. The thickness at all four starting point remains the same. A breakdown of all four starting tow orientation values is given in Table 6.5 Where U(−30, 30) is a vector with each component taking on a uniformly random value from−30◦ to30◦. An offset of ±15◦is added to random cases 2 and 3, to gauge the effect that varying the initial average direction of the of the tow paths has on the optimization result.
Table 6.5: Randomized starting points
Design variable Nominal case Random case 1 Random case 2 Random case 3
Figure 6.12: Result of structural optimization of uCRM-9: unsteered (left) vs. tow-steered (right)
The initial and optimized thickness and main tow path patterns for the upper and lower skins and corresponding final masses for all four cases are shown in Figure 6.13. From these results it can be seen that all four starting points converge to roughly the same final structural weight, with the largest difference between cases being only 0.17%. Not only do the overall masses match between the four designs, but distribution of mass, as shown by the panel thickness contours, are indistinguishable. Finally, by comparing the main tow patterns for the upper and lower skins of each design, we see that the tow paths are nearly identical as well. The only location where slight difference can be observed are at the locations of tips. For each case, in these regions the tow paths seem to remain at the at initial orientation defined in the region at the starting point. Again, this is likely due to the low design sensitivities in this region of the wing, leading to relatively flat design space as the optimizer attempts to converge these regions.
From these results, it seems safe to conclude that the presence of local minima is not a problem for the tow-steered wingbox parametrization considered in this work. To be rigorous, it should be noted that by the periodic nature of the angular variables used to define the tow-paths (i.e. θ = θ + n2π) there will be multiple solutions that represent exactly the same design. However, as long as these are the only minima that occur, we can restrict the optimization to a subset of the periodic domain (i.e. 0 ≤ θ ≤ 2π) and ensure that the minima located is likely globally optimal. Of course there still exists the possibility of local minima in locations of the design space that may have been missed by this random sampling of the design space. In general, it is difficult to guarantee global convexity of general optimization problems. Instead, we will use this case study to conclude that the design optimization appears to be convex in a large region of the design space. Finally, the conclusion of this study should not be used to disprove the concern of local minima introduced by the chosen parametrization for all structural design problems, but merely for the wingbox case studies considered in the remainder of this work.
Figure 6.13: The resulting main pattern and panel thickness distribution from each starting point
CHAPTER 7
Fuel burn optimization
In the previous chapter, a series of structural optimization problems were solved to gain some insight into the structural benefits of tow-steered composite optimization. In this chapter, the insights will be extended to the aeroelastic benefits specific to wing design. To this end, a series of fuel burn minimization problems will be performed on the uCRM-9. To quantify the benefits, the optimizations are run with both a tow-steered and fixed ply orientation conventional composite wingbox design. The benefits of tow-steering will then be reassessed with a conventional design where the plies of the wing skins are free to be rotated by the optimizer. Next, the performance penalty associated with the manufactur- ing constraints introduced in Chapter5will be quantified by re-optimizing the tow-steered design without them. The tow-steered and fixed ply orientation conventional composite design optimizations will then be performed on the uCRM-13.5. Including this design pro- vides insight into the benefits of tow steering with respect to high-aspect-ratio wing design. Finally, the effect of aspect ratio on the wing design performance will be refined. This will be accomplished by re-optimizing the tow-steered and fixed ply orientation conventional composite designs on the uCRM model for a wing aspect ratio of 7.5, 9, 10.5, 12, and 13.5 and analyzing the resulting trends in performance.