# Multi-Fidelity Gradient-Based Optimization for High-Dimensional Aeroelastic Configurations
## 2 Modeling and Analysis
### Accuracy and Cost of Derivative Calculations  
The adjoint-based process for computing design derivatives was verified by comparison of analytically computed derivatives to first-order-accurate, forward finite differences for the baseline cruise case. The lift coefficient was selected as the objective, and derivatives with respect to the Yehudi twist angle and the thickness of an arbitrarily selected structural element, \(t_{42}\) (the 4 th trailing edge spur outboard from the wing-fuselage junction), were compared. Step sizes of 0.01 and 0.0001 were selected for the shape and sizing variables; to settle on these values, increasingly small step sizes were tested until the gradients roughly matched. Table 2 shows the resulting lift coefficient derivatives (an analytic derivative is not available for the low-fidelity rigid case). While optimizations regenerate the aerodynamic grids for each new design, the aerodynamic grids here were modified by smoothly mapping the initial mesh onto the perturbed geometry. For aeroelastic cases, the perturbed structural grids were generated by rebuilding and remeshing the IML geometry. For both models, gradients with respect to structural thicknesses (as well as angles of attack) were observed to be more precise when compared to shape derivatives for the flexible vehicle. Because the shape derivative was found to be more accurate for the rigid case, the increased error for the flexible case may in part be due to the neglect of structural design velocities. To help estimate how large this source of error may be, the HF aeroelastic model was also perturbed while fixing the IML's geometry and mesh. This resulted in a finite-difference gradient of 0.05409046, yielding an adjoint error of \(-0.0768\%\), thus suggesting the structural shape gradient term in Equation (6) changes the gradient by around \(-0.2796\%\). However, note that a mainly structural response such as an aggregated stress constraint may have a larger discrepancy when compared to an aerodynamic load such as the lift coefficient. For the LF model, imprecise geometric sensitivities are likely a larger contributing factor. Nonetheless, the analytic derivatives compared reasonably well to finite differences. The computational efficiency of computing derivatives is next assessed for LF and HF analyses using parameterizations of differing dimensionality. It was observed for the most computationally demanding analysis case, HF aeroelastic analysis, that computational \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Variable** & **Model** & **Adjoint** &
\begin{tabular}{c} **Finite** \\ **Difference** \\ \end{tabular} & **Difference (\%)** \\ \hline \multirow{4}{*}{Yehudi twist} & Low-fidelity aeroelastic & 0.04026 & 0.03796 & 6.06 \\ \cline{2-5}  & High-fidelity aeroelastic & 0.05405 & 0.05394 & 0.203 \\ \cline{2-5}  & High-fidelity rigid & 0.07958 & 0.07954 & 0.052 \\ \hline \multirow{4}{*}{\(t_{42}\)} & Low-fidelity aeroelastic & 0.06436 & 0.06431 & 0.088 \\ \cline{2-5}  & High-fidelity aeroelastic & 0.07160 & 0.07149 & 0.157 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Derivatives of the lift coefficient with respect to shape and thickness design variables. Finite difference results are forward differenced with a relative step size of 0.01 for the shape variable and 0.0001 for the structural variable. Figure 6: HF adjoint, geometric sensitivities, and their dot product around the baseline aeroelastic solution. **(a)** Log-scale lift coefficient sensitivity. **(b)** Log-scale Yehudi twist sensitivity. time scales slowly with the size of the design space. Results were measured in terms of wall-clock time. Each HF run utilized 32 CPUs and involved 600 steps (or reaching a consistent residual of \(1\times 10^{-10}\) for rigid runs); parallelism was exploited where possible. The LF analysis typically involved 15 steps, but in some cases required 30 steps to achieve convergence. Although the LF aeroelastic model is capable of parallel execution, 1 CPU was used.