# Multi-Fidelity Gradient-Based Optimization for High-Dimensional Aeroelastic Configurations
## 3 Results
### Summary and Comparison
The multi-fidelity optimizer was able to achieve most of the objective reduction within the first several high-fidelity evaluations. In the rigid P1a case (7-DoF Kulfan), after accepting the first design step, the multi-fidelity optimizer rejected the second step as the corrected low-fidelity model led to a design that increased rather than decreased the high-fidelity objective. Consequently, the trust region was halved. (The initial trust region covered the entire design domain.) The optimizer accepted the majority of subsequent steps, though the trust region steadily decreased to a size near 0.0625-0.125 (in the normalized design space \(\mathbf{D}_{i}\in\{0,1\}\)). This resulted from the expected improvement ratio between the multi-fidelity and high-fidelity objectives being near enough to one to indicate that the corrected low-fidelity model was sufficient to make progress, but not accurate enough to warrant trusting over a larger space. However, after about 25 high-fidelity evaluations, the search entered a region of the design space where the multi-fidelity model was not accurate, and the optimizer rejected a series of design steps that increased the high-fidelity objective and decreased the trust region in an attempt to find a step size where the multi-fidelity model was accurate. Of the airfoil design cases, the aeroelastic case P1a exhibited the least benefit from the multi-fidelity approach. For the first ten evaluations, the optimizer once again found the multi-fidelity model to be sufficiently accurate and maintained the trust region over the entire design domain. After that point, the multi-fidelity model did not accurately predict the high-fidelity objective, and the trust region was quickly reduced. Once the trust region had fallen below 0.001 for three consecutive iterations (a selectable parameter), the optimization reverted to single-fidelity after approximately 25 evaluations, making use of the approximate Hessian already built using high-fidelity information. Despite reverting  
Figure 17: Performance of multi-fidelity model within airfoil design. (**a**) Rigid 7-DoF Kulfan; (**b**) Aeroelastic 7-DoF Kulfan. to high-fidelity optimization, there was no discernible reduction in the objective function, suggesting that the multi-fidelity optimizer approximately found the minimum. For reference, Table 10 lists the optimization costs for the cases with sizing. As discussed previously, the results were generally mixed, with multi-fidelity at times providing some cost savings, but often times not. Interestingly, while the cases could likely be converged more deeply, both optimizers appeared to reach a 99% reduction within 20 high-fidelity evaluations. This is likely dependent on the scale factors of the penalty functions, as the objective is generally around 0.6-2.0, with the penalty functions increasing the penalty-constrained objective by one to two orders of magnitude. Thus, much of the penalty-constrained objective reduction was achieved simply by locating a feasible design space, rather than simply reducing the fuel burn objective. Figure 18 shows the behavior of the multi-fidelity optimizer for combined structural sizing and shape optimization. Compared to shape optimization alone (rigid and aeroelastic without sizing), the multi-fidelity optimization provided an inconsistent benefit in the 17-DoF airfoil case (P3b) and a cost penalty in the 7-DoF airfoil case (P3a). In case P3a, the multi-fidelity model had acceptable quality to maintain the trust region around 1.0 for approximately 15 evaluations, briefly reducing to 0.25. Then, the trust region was steadily reduced, though a few points were rejected due to increasing the objective function. After approximately 25 evaluations, most of the design steps were rejected and the multi-fidelity optimizer made no significant progress. While case P3b rejected fewer design steps in the first 15 evaluations, the trust region grew steadily smaller as the optimizer rejected almost all subsequent steps.