# Multi-Fidelity Gradient-Based Optimization for High-Dimensional Aeroelastic Configurations
## 3 Results
### Summary and Comparison
For reference, Figure 14 illustrates the baseline and optimized airfoil shapes for all cases including shape variables, where undeflected jig shapes are shown for the aeroelastic cases. In particular, the top and bottom rows show undeflected airfoil shapes at the Yehudi and tip sections, respectively. Columns from left to right show results for rigid, aeroelastic shape-only, and aeroelastic with sizing cases. The rigid Yehudi geometries were largely unchanged, although minor shape changes were likely needed to reduce the shock strength. Rigid tip sections varied significantly in twist angle, however. This is to be expected, as aeroelastic deflection tends to cause downward wing twist, so with deflection, optimized rigid and aeroelastic geometries are likely similar in the twist angle. Visually, the aeroelastic optima with and without sizing are difficult to compare; however, both visually and in terms of final fuel burn, a higher  
Figure 14: Baseline and optimized airfoil shapes, where aeroelastic cases show the undeflected jig shapes. (**a**) Yehudi, rigid; (**b**) Yehudi, aeroelastic; (**c**) Yehudi, aeroelastic with sizing; (**d**) Tip, rigid; (**e**) Tip, aeroelastic; (**f**) Tip, aeroelastic with sizing. -dimensional airfoil section (or an even higher one) appears to be necessary to obtain the \"true\" optimized geometry. For reference, Figures 15 and 16 show a sample solution to P3a, which yielded the lowest fuel burn among the shape and sizing cases. In particular, Figure 15 shows the undeflected geometry, as well as the deflected shapes during the cruise and maneuver scenarios. Shown also for reference are the trimmed baseline geometries, which had significantly larger deflections due to the lower aerodynamic efficiency, thus requiring a larger fuel mass to achieve the prescribed cruise range (and more lift to meet load factor constraints, thus leading to more induced and likely wave drag). Figure 16a shows the thickness distributions of the structure, with the top, middle, and bottom showing top skins, ribs and spars, and bottom skins, respectively. Figure 16b shows the corresponding stress contours (von Mises stress normalized by yield stress) in the maneuver scenario. For reference, the stress constraint used in the optimization enforces the normalized stress to not exceed \(2/3\), which is the inverse of the factor of safety (1.5). To achieve minimum structural mass while remaining feasible, the optimal stress contours would yield a maximum value of \(2/3\) in each structural component. This appears to be the main driver for much of the wing skins, although a handful of skins still appeared thicker than they needed to be. Elsewhere in the wing, the aerodynamic performance and trim constraints (as well as the KS aggregation) appeared to be influencing the thickness  
Figure 16: Stress and thickness contours of the best shape+sizing optimization solution (P3a). (a) Structural thicknesses; (b) von Mises over yield stress for a +2.5 g maneuver. Figure 15: Baseline and optimized jig shapes and deflected geometries in 1g cruise and 2.5 g pull-up conditions (7-DoF airfoil parameterization). distributions as well. For instance, while the tip skin thicknesses seemed to be approaching the lower bound of 0.005 m, the tip rib and spar thicknesses remained relatively thick in spite of the low stresses in the faces. This could mean that more torsional or bending stiffness was needed to ensure an adequate lift or moment. In general, the stress and thickness contours compared reasonably well to those of [PERSON] et al. (2016), although the thicknesses were somewhat larger, leading to a somewhat heavier structure. Aside from optimization performance, one potential cause is the lack of stiffeners modeled in the structure, which may allow for thinner ribs and spars. The skins that intersect the wingbox trailing edge and wing-fuselage junction ribs were also quite thick and may also benefit from modeling stiffeners. There also appeared to be a stress concentration at the Yehudi on the bottom surface.