# A Study on the Surrogate-Based Optimization of Flexible Wings Considering a Flutter Constraint +
###### Abstract  
Accounting for aeroelastic phenomena, such as flutter, in the conceptual design phase is becoming more important as the trend toward increasing the wing aspect ratio forges ahead. However, this task is computationally expensive, especially when utilizing high-fidelity simulations and numerical optimization. Thus, the development of efficient computational strategies is necessary. With this goal in mind, this work proposes a surrogate-based optimization (SBO) methodology for wing design using a predefined machine learning model. For this purpose, a custom-made Python framework was built based on different open-source codes. The test subject was the classical Goland wing, parameterized to allow for SBO. The process consists of employing a Latin Hypercube Sampling plan and subsequently simulating the resulting wing on SHARPy to generate a dataset. A regression-based machine learning model is then used to build surrogate models for lift and drag coefficients, structural mass, and flutter speed. Finally, after validating the surrogate model, a multi-objective optimization problem aiming to maximize the lift-to-drag ratio and minimize the structural mass is solved through NSGA-II, considering a flutter constraint. This SBO methodology was successfully tested, reaching reductions of three orders of magnitude in the optimization computational time.  
multidisciplinary design optimization; aeroelasticity; multi-objective optimization; wing design; surrogate models +
Footnote †: This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license ([[https://creativecommons.org/licenses/by/4.0/](https://creativecommons.org/licenses/by/4.0/)]([https://creativecommons.org/licenses/by/4.0/](https://creativecommons.org/licenses/by/4.0/))).  
problems were aimed at maximizing range [8; 9; 11], although fuel burn [12] and structural mass [10] minimization has also been chosen.  
Even though reduced-order and surrogate models have been applied to aeroelasticity problems for some time [13], especially to study unsteady nonlinear aeroelasticity problems that arise from unconventional features such as limit-cycle oscillations [14; 15; 16], their usage for MDO problems considering flutter is scarce [17; 18; 19] in the open literature, particularly for MDO problems considering aerodynamic performance and structural weight. For instance, [PERSON] et al. [17] developed an MDO strategy that uses multi-fidelity solvers and surrogate models to design strut-braced and high-aspect-ratio wings considering flutter and stress constraints. They found that, although the flutter constraint was respected in the optimization process, unaccounted buckling was observed in a nonlinear assessment of the optimized strut-braced wing aircraft design. [PERSON] and [PERSON] [18] implemented an optimization framework based on ROMs that couples different open-source codes, including SHARPy, for minimizing the mass of a flexible strut-braced wing while accounting for flutter speed as a constraint. [PERSON] and [PERSON] [19] developed a methodology combining in-house codes with surrogate models to optimize the structural layout of a conventional aircraft such that its mass is minimized while considering stress and flutter constraints. The application of surrogate models based on machine learning is scarcer. Nevertheless, this research field is starting to emerge, particularly for aerodynamics [6; 20].  
In this work, a surrogate-based optimization (SBO) strategy is presented to conceptually design flexible wings based on machine learning-based surrogate models with the aim of maximizing aerodynamic performance, minimizing structural mass, and satisfying the flutter constraint.