# Multi-Fidelity Gradient-Based Optimization for High-Dimensional Aeroelastic Configurations
## 3 Results
### Summary and Comparison
\begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Problem** & **FB\({}_{0}\) (kg)** & **FB\({}^{*}\) (kg)** & **Initial Constraints** & **Final Constraints** \\ \hline \multirow{3}{*}{P1a} & \multirow{3}{*}{109,846.1} & \multirow{3}{*}{65,360.0 (\(-\)40.50\%)} & \(7.8224\times 10^{-1}\) & \(1.4390\times 10^{-2}\) \\  & & & \(-1.8457\times 10^{-2}\) & \(-2.0931\times 10^{-2}\) \\  & & & \(-1.5429\times 10^{-3}\) & \(9.3123\times 10^{-4}\) \\ \hline \multirow{3}{*}{P1b} & \multirow{3}{*}{77,726.8} & \multirow{3}{*}{60,008.4 (\(-\)22.80\%)} & \(6.4153\times 10^{-1}\) & \(9.9342\times 10^{-3}\) \\  & & & \(-2.0298\times 10^{-2}\) & \(-2.9444\times 10^{-2}\) \\  & & & \(2.9824\times 10^{-4}\) & \(9.4445\times 10^{-3}\) \\ \hline \multirow{3}{*}{P2a} & \multirow{3}{*}{109,846.1} & \multirow{3}{*}{93,845.3 (\(-\)14.57\%)} & \(7.8224\times 10^{-1}\) & \(6.8507\times 10^{-2}\) \\  & & & \(-1.8457\times 10^{-2}\) & \(-1.7051\times 10^{-2}\) \\  & & & \(-1.5429\times 10^{-3}\) & \(-2.9491\times 10^{-3}\) \\  & & & \(1.3408\times 10^{0}\) & \(6.7895\times 10^{-3}\) \\  & & & \(2.6601\times 10^{-1}\) & \(1.6474\times 10^{-2}\) \\ \hline \multirow{3}{*}{P2b} & \multirow{3}{*}{77,726.8} & \multirow{3}{*}{67,303.7 (\(-\)13.41\%)} & \(6.4153\times 10^{-1}\) & \(2.0281\times 10^{-2}\) \\  & & & \(-2.0298\times 10^{-2}\) & \(-1.7018\times 10^{-2}\) \\  & & & \(2.9824\times 10^{-4}\) & \(-2.9823\times 10^{-3}\) \\  & & & \(1.0383\times 10^{0}\) & \(6.1785\times 10^{-4}\) \\  & & & \(5.2509\times 10^{-1}\) & \(4.3878\times 10^{-3}\) \\ \hline \multirow{3}{*}{P3a} & \multirow{3}{*}{109,846.1} & \multirow{3}{*}{62,179.5 (\(-\)43.39\%)} & \(7.8224\times 10^{-1}\) & \(1.1984\times 10^{-2}\) \\  & & & \(-1.8457\times 10^{-2}\) & \(-2.3495\times 10^{-2}\) \\ \cline{1-1}  & & & \(-1.5429\times 10^{-3}\) & \(3.4952\times 10^{-3}\) \\ \cline{1-1}  & & & \(1.3408\times 10^{0}\) & \(-4.6511\times 10^{-3}\) \\ \cline{1-1}  & & & \(2.6601\times 10^{-1}\) & \(-4.1505\times 10^{-3}\) \\ \hline \multirow{3}{*}{P3b} & \multirow{3}{*}{77,726.8} & \multirow{3}{*}{72,152.8 (\(-\)7.17\%)} & \(6.4153\times 10^{-1}\) & \(4.5460\times 10^{-2}\) \\  & & & \(-2.0298\times 10^{-2}\) & \(-3.8832\times 10^{-2}\) \\ \cline{1-1}  & & & \(2.9824\times 10^{-4}\) & \(1.8832\times 10^{-2}\) \\ \cline{1-1}  & & & \(1.0383\times 10^{0}\) & \(2.6112\times 10^{-3}\) \\ \cline{1-1}  & & & \(5.2509\times 10^{-1}\) & \(5.9354\times 10^{-2}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 8: Summary of aeroelastic optimization cases. the penalty-constrained SLSQP, which is perhaps a fairer comparison to the BFGS-based optimizers, yielded a fuel burn of 72,519.3 kg with final constraints of \([2.4784\times 10^{-2},\)\(-1.3242\times 10^{-2},\)\(-6.7584\times 10^{-3}]^{\mathrm{T}}\), an 11% difference. The same trend was observed to a greater extent for the sizing-only cases, with the higher dimensional airfoil producing a significantly lower fuel burn (although it is possible the P2a result stalled). Among simultaneous shape and sizing cases (P3a and P3b), the same trend was unfortunately not observed. While P3a yielded a large decrease in fuel burn from P2a, the solution to P3b was in fact worse than P2b (same problem, but lacking shape design variables), as well as P3a (same problem, but with lower-dimensional airfoils). This seems to indicate either a local minimum or, perhaps more likely, simply a difficult problem for the optimizers to solve. Potentially, numerical noise from remeshing, as well as neglecting structural grid velocities (which may be important for the stress gradients) could be contributing factors. Nonetheless, all cases were at least successful in approximately satisfying the constraints and reducing fuel burn compared to the baseline geometry. Additionally, the single- and multi-fidelity all yielded roughly the same solution, with the exception of rigid P1b, where the multi-fidelity optimizer found a slightly worse solution.