# Multi-Fidelity Gradient-Based Optimization for High-Dimensional Aeroelastic Configurations
## 4 Discussion  
The definition and implementation of a multi-fidelity, aero-structural analysis capability were described, providing the foundation for a family of benchmark problems in multi-fidelity design optimization. The models provide a rich set of selectable features, including physics coupling, data sources, parameter sets (e.g., aerodynamic shape versus structural sizing), and scalability of the design domain size. A key aspect of this work is the efficient computation of derivatives with respect to the design variables. This is a critical enabler of large-scale design optimization in both single- and multi-fidelity settings. These sensitivities were used to demonstrate gradient-based, multi-fidelity optimization for design problems that are infeasible using typical, surrogate-based multi-fidelity methods. The design domains spanned tens to hundreds of design variables, considering aerodynamic shape and structural sizing separately and in combination. While the MF-BFGS approach was capable of performing design optimization with hundreds of variables, the scalability of the method was confounded with the consideration of different problem types. The introduction of structural sizing with hundreds of design variables required additional design constraints and gradient calculations for variables not present in the shape optimization with tens of variables. Thus, impacts on optimizer performance due to design domain dimension versus the character of the design problem could not be separated. Considering shape variables alone, MF-BFGS provides a benefit over single-fidelity BFGS in terms of objective-function reduction versus the number of high-fidelity function evaluations, whether rigid or aeroelastic. However, cases with structural sizing variables (aeroelastic only) performed less consistently. Using Kulfan airfoils with 17 degrees of freedom, the multi-fidelity optimizer produced a 45-55 percentage points greater objective function reduction than single-fidelity optimization in the first few evaluations. However, the gains were temporary, and both optimizers performed similarly in later iterations. In contrast, the seven-degree-of-freedom Kulfan cases tended to perform worse using the multi-fidelity approach in the first several iterations. This trend of the 17-degree-of-freedom airfoil exhibiting better multi-fidelity performance than the 7-degree-of-freedom airfoil was also true of the aeroelastic shape design, but not of the rigid shape design. Several aspects could make the structural design problem more challenging than aerodynamic shape optimization. First, the sizing problem requires two additional constraints for the maneuver load factor and aggregated stress constraint. Adding these constraints to the objective as quadratic penalty terms could potentially make the design space more difficult to navigate, and the constraint penalty and KS aggregation functions require tunable parameters. The parameter selection dictates a compromise between design space curvature and constraint conservatism, which could have a significant impact on optimizer  
Figure 18: Performance of multi-fidelity model within simultaneous shape and structural design. (a) 7 DoF Kulfan; (b) 17 DoF Kulfan. performance. Second, the aggregation of stress constraints combined with structural mesh regeneration may lead to a noisy response that slows convergence. The computation of structural shape sensitivities and the development of smoothly mapping meshes to perturbed geometry are current areas of development. Third, the external finite-differencing of the VLM mesh generation process for low-fidelity gradients and the neglect of structural design velocities (i.e., sensitivity of the structural mesh to geometry changes) could lead to inaccuracies that cause the multi-fidelity correction to suffer and produce sub-optimal line search results.