# Multi-Fidelity Gradient-Based Optimization for High-Dimensional Aeroelastic Configurations
## 2 Modeling and Analysis
### Rigid and Aeroelastic Analysis with Design Sensitivities
#### 2.2.1 Rigid Aerodynamic Analysis  
Aerodynamic solutions of the rigid uCRM configuration in steady, transonic flight are computed at two fidelity levels, as outlined in Table 1. The LF solution is computed with a widely distributed, linear, VLM methodology known as Athena Vortex Lattice (AVL), developed by [PERSON] and [PERSON] and [PERSON] (2016). Drag estimates produced by AVL are corrected for wave drag using the Korn equation, applied in stripwise fashion using an aerodynamic technology factor of 0.87 ([PERSON] and [PERSON], 2017). Since AVL is not analytically differentiated, the sensitivities of the computed aerodynamic quantities with respect to the design parameters are computed with one-sided finite differences using an appropriately chosen step size (a relatively large normalized step size of 0.1, on account of AVL's low decimal precision). While the cost of computing these sensitivities grows linearly with the number of design variables, the overall cost is small owing to the speed of LF analysis and is not considered a bottleneck for the configuration studied herein. Furthermore, for aeroelastic analysis, a different VLM formulation is used that is analytically differentiated. Higher-fidelity aerodynamic solutions are computed with FUN3D ([PERSON], 2017), which can solve the steady or unsteady Euler or Navier-Stokes equations on unstructured meshes. The benefit of FUN3D over other available solvers is that flow equations are analytically differentiated internally using the discrete adjoint approach, which permits the scalable and accurate computation of design derivatives in a manner consistent with the discretized governing equations. An arbitrary design objective (or constraint function) \(f\equiv f(\mathbf{D},\mathbf{Q_{a}},\mathbf{X_{g}})\) is assumed to depend on the set of design variables, \(\mathbf{D}\equiv(\mathbf{x_{s}},\mathbf{x_{a}})^{T}\), the aerodynamic state variables, \(\mathbf{Q_{a}}\), and the computational mesh, \(\mathbf{X_{g}}\). For gradient-based optimization, the variation of \(\mathcal{J}\) with \(\mathbf{D}\) is required, where \(\mathbf{Q_{a}}\) and \(\mathbf{X_{g}}\) are dependent on \(\mathbf{D}\). Following the adjoint approach of [PERSON] and [PERSON] [PERSON] and [PERSON] (1983), the total derivative of \(f\) at a known rigid solution is expressed as  
\[\frac{d\mathcal{J}}{d\mathbf{D}}=\frac{\partial f}{\partial\mathbf{D}}+{ \mathbf{A_{a}}}^{T}\frac{\partial\mathbf{R_{a}}}{\partial\mathbf{D}}-{ \mathbf{A_{g}}}^{T}\bigg{(}\frac{\partial\mathbf{X_{g}}}{\partial\mathbf{D}} \bigg{)}_{\text{surface}}, \tag{1}\] \begin{table}
\begin{tabular}{c c c} \hline \hline \multirow{2}{*}{**Response**} & \multicolumn{2}{c}{**Fidelities and Solvers**} \\ \cline{2-3}  & **Low-Fidelity (LF)** & **High-Fidelity (HF)** \\ \hline Rigid & AVL ([PERSON] and Youngren, 2016) + Korn wave drag * & FUN3D ([PERSON], 2017) Euler \\ \hline Aeroelastic & VLM + TACS ([PERSON] and Youngren, 2016) & FUN3D ([PERSON], 2017) Euler + TACS (Drela and Youngren, 2017) \\  & via MPHYS ([PERSON] and Youngren, 2017) & via FUNtoFEM (Drela and Youngren, 2017; Drela and Youngren, 2017) \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Modeling fidelity by response type. (“\({}^{*}\)” denotes gradient computation through forward differencing of tool chain.