# Multi-Fidelity Gradient-Based Optimization for High-Dimensional Aeroelastic Configurations
## 1 Introduction  
The simultaneous optimization of aircraft shape and internal structural size for transonic flight is excessively costly and a barrier to the design of efficient and effective aircraft. Optimization cost scales poorly with the number of design parameters for many kinds of algorithms, rendering design impractical when many parameters are needed, e.g., to finely tailor wings aerodynamically and structurally for improved performance. Practical design optimization of high-fidelity, physics-based models in such high-dimensional spaces generally involves three ingredients: (1) the use of local, gradient-based optimization to avoid excessive function evaluations, (2) the use of analytic differentiation and adjoint variables to mitigate cost growth in the derivative calculation, and (3) the avoidance of cheaper-to-analyze surrogate models owing to the high cost of constructing such surrogates in high-dimensional spaces. This paper addresses methodological contributions impacting each of these ingredients. The first contribution relates to the analytic calculation of design derivatives with adjoints, as parameterized by an analytical geometry description. The use of adjointvariables to compute derivatives of aerodynamic problems, and aeroelastic problems to a lesser extent, has been well studied [1; 2]. The cost of computing derivatives with these variables has been seen to grow only very slowly with design space dimensionality. [PERSON] and coworkers [3; 4; 5] developed aeroelastic design optimization frameworks for inviscid and viscous aerodynamics built on a custom analytic geometry, a vision that is extended here with a general-purpose, design model (CAPS) [6] and geometry engine (ESP) [7] framework supportive of multi-fidelity modeling [8]. This framework leverages two modern, fluid-structure-coupling procedures, FUNtoFEM [9; 10] and MPHYS [11]. This paper verifies the design sensitivities produced by this framework for linear and nonlinear aerodynamics and assesses the scalability of their calculation with respect to design complexity. With ESP and CAPS being built on a true CAD kernel (OpenCASCADE [12]) and CAPS having automatic aerodynamic and structural meshing capabilities, the choice of software tools was motivated by the ability to compute sensitivities with respect to geometric parameters that are meaningful to designers (e.g., twist or airfoil thickness), as opposed to mesh-based shape parameterizations in the literature. At the same time, however, the approach also introduces challenges such as those associated with rebuilding geometries or regenerating unstructured grids throughout an optimization. The second contribution is the quantification of the scalability of the multi-fidelity Broyden-Fletcher-Goldfarb-Shannon (MF-BFGS) algorithm developed by [PERSON] and [PERSON] [13; 14]. As opposed to typical multi-fidelity optimization approaches optimizing over a sequence of surrogate models (of potentially high dimension), this algorithm leverages high-fidelity gradient information to determine an objective-reducing search direction and then utilizes a less expensive multi-fidelity model to reduce analysis costs in the line search. In a study of several analytical functions with up to 25 design variables [14], [PERSON] and [PERSON] found MF-BFGS to scale better than multi-fidelity trust region model management (TRMM) [15; 16; 17]. [PERSON] confirmed that the MF-BFGS method generally out-performed BFGS for an aeroelastic model of a tailless lambda-wing vehicle [13], but noted that the lack of analytic derivatives in that work made some optimization results difficult to interpret. As this model was also limited in complexity to seven design variables [13] (four aerodynamic and three structural), the authors in this paper sought a more richly parameterized and representative model, as well as analytical derivatives with which to assess MF-BFGS scalability.