# Multi-Fidelity Gradient-Based Optimization for High-Dimensional Aeroelastic Configurations
## 3 Results
### Sample Optimization and Consideration of Constraint Handling  
To first demonstrate how a typical optimizer might perform for the particular model and design problem, problem P1a was solved using the high- and low-fidelity aeroelastic models. The SLSQP algorithm in OpenMDA's SciPy wrapper was used as an optimizer. Figure 9 shows the convergence histories of the objective function and maximum constraint value. The low-fidelity run required 8 gradient calls and 23 non-gradient calls. However, the final load factor constraint was not active as expected, which could indicate issues with gradient accuracy or possibly numerical noise (which, to some extent, would also appear in the high-fidelity design space due to the regeneration of unstructured meshes). The high-fidelity run used 23 gradient calls and 62 non-gradient calls. Interestingly, the low-fidelity solution yielded a somewhat larger fuel burn (84,747 kg compared to 64,016), which more closely matches the solution reported by [PERSON] et al. (2018) (84,072 kg). Presumably, a RANS solution would more closely match that of [PERSON] and the VLM method, due to the presence of viscous drag. Because the BFGS-based optimizer considered below does not handle inequality constraints directly and thus relies on penalty constraints, the high-fidelity SLSQP case was also run using the same penalty constraint formulation, taking 20 gradient calls and 53 non-gradient calls. Unsurprisingly, the case ultimately required more iterations to converge, with the objective converging much more slowly, although the feasible design space was located more quickly. Unfortunately, the final solution found did not match that of the direct constraint handling case. This could be due to noise from remeshing or simply indicate a complex design space with local minima or very flat regions. \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Case** & **DVs** & **Airfoil DoF** & **\# DVs** & **Constraint Equations** \\ \hline P1a & \(\{\mathbf{x_{s_{a}}},\mathbf{x_{s_{t}}},\mathbf{x_{a}}\}\) & 7 & 17 & (21)–(23) \\ P1b & \(\{\mathbf{x_{s_{t}}},\mathbf{x_{s_{t}}},\mathbf{x_{a}}\}\) & 17 & 37 & (21)–(23) \\ P2a & \(\{\mathbf{x_{t}},\mathbf{x_{a}}\}\) & 7 & 285 & (21)–(25) \\ P2b & \(\{\mathbf{x_{t}},\mathbf{x_{a}}\}\) & 17 & 285 & (21)–(25) \\ P3a & \(\{\mathbf{x_{s_{a}}},\mathbf{x_{s_{t}}},\mathbf{x_{t}},\mathbf{x_{a}}\}\) & 7 & 301 & (21)–(25) \\ P3b & \(\{\mathbf{x_{s_{a}}},\mathbf{x_{s_{t}}},\mathbf{x_{t}},\mathbf{x_{a}}\}\) & 17 & 321 & (21)–(25) \\ \hline \hline \end{tabular}
\end{table} 
Table 6: Optimization problem labels. Problems 2 and 3 are only solved with aeroelastic models rather than rigid as well, as they include structural sizing and stress constraints. \begin{table}
\begin{tabular}{c c c c c} \hline \hline  & \multicolumn{2}{c}{**Shape**} & \multirow{2}{*}{**Structural**} & \multirow{2}{*}{**Aerodynamic**} \\ \cline{2-2} \cline{4-5}  & **Airfoil** & & & \\ \hline Vector name & \(\mathbf{x_{s_{a}}}\) & \(\mathbf{x_{s_{t}}}\) & \(\mathbf{x_{t}}\) & \(\mathbf{x_{a}}\) \\ \hline Dimensionality & 14 (7 DoF) or 34 (17 DoF) & 2 & 283 & 1 or 2 \\ \hline \hline \end{tabular}
\end{table} 
Table 5: Design variable designations for the different parameterizations. For reference, Figure 10 shows the resulting airfoil shapes at the Yehudi and tip wing sections. At the tip section, the baseline and directly constrained high-fidelity results appear to be quite similar, while larger differences can be seen in the low-fidelity and penalty-constrained high-fidelity results. While the low-fidelity result (which has higher twist and less camber) could simply be due to the lower fidelity physics (in combination with moment constraints, which would effectively influence chordwise and spanwise lift distributions), the penalty-constrained high-fidelity result (which has significantly different trailing edge camber) may be due more to numerical difficulties introduced by the penalty formulation and weights.