The usage of numerical optimization algorithms has allowed for lighter and more aerodynamically efficient wing designs. These algorithms include, among others, Aerodynamic Shape Optimization (ASO) and Topology Optimization (TO). In the former, a given shape, for example a wing or an airfoil, after being adequately parameterized is optimized for a specific aerodynamic goal, such as lift-to-drag ratio maximization or drag minimization, while fulfilling a prescribed set of constraints. The latter consists in finding an internal structure that minimizes its compliance in a given solid domain for a known set of boundary conditions. ASO [1
] and TO [2
] have been applied in both academic and industrial applications related to aeronautics, including wings and airfoils.
TO is a useful computational tool that can be used to reach lighter wing structures and so it has been applied to different aircraft types such as transport, unmanned aerial vehicles and even micro air vehicles in recent decades. Maute and Allen [3
] developed a framework to optimize the topology of a wing structure considering Fluid-Structure Interaction (FSI). Krog et al. [4
] in a joint effort between Airbus and Altair tested different topology optimization strategies considering 10 load cases to optimize a wing-box rib. Gomes and Suleman [5
] used TO for the design of a wing-box such that the aileron reversal speed is maximized. Stanford and Ifju [6
] used aeroelastic TO to design the membrane structure of a micro air vehicle’s wing, considering different objective functions including lift-to-drag ratio maximization. In [7
], the authors tested their TO methodology on the internal structure of a tapered and swept wing-box, formed by a constant symmetrical airfoil, subjected to a pressure load. They observed an asymmetrical structure, formed to withstand the also asymmetrical span-wise pressure load, which presents more material near the root to support the bending and cross trusses to withstand shear. In [8
], the internal structure of a unmanned aerial vehicle’s wing is optimized recurring to TO considering aerodynamic loads. Dunning et al. [9
] optimized the wing of the Common Research Model (CRM) considering the trimmed lift. Félix et al. [10
] considered the wing’s self-weight, besides an aerodynamic load, to design it topologically. Capasso et al. [11
] employed TO to design a morphing structure made of a hyperelastic material.
ASO of airfoils and wings can be conducted by means of a gradient-based optimization algorithm or a gradient-free one. The former is more common than the latter and it is particularly efficient when gradients are computed by means of the adjoint method [12
] to explore high-dimensional design spaces and find local minima [13
]; however, global search gradient-free algorithms (such as genetic algorithms) might be more adequate to find a global minimum. Nemec et al. [14
] performed ASO of the RAE2822 airfoil for two objectives and multiple flow conditions, using a gradient-based algorithm where the sensitivities are computed by means of the adjoint method. The results obtained compared well with those reached with a genetic algorithm. Later, Zingg et al. [15
] compare these two optimization approaches for ASO of airfoils, reaching the conclusion that despite both reached the same solution, the latter one takes 5 to 200 times longer than the former. Martins and his coworkers have been conducting ASO using gradient-based optimizations for several wing applications, spanning from conventional aircraft configurations [16
] to unconventional solutions (e.g., camber morphing [17
]) and designs (the blended-wing-body [18
] and the strut-braced wing [19
]) with the aim of improving performance. Several parameterization techniques are compared for ASO of airfoils in [20
], including Free-Form Deformation (FFD), non-uniform rational B-Splines (NURBS), Class Function/Shape Function Transformations (CST) and Hicks-Henne. The authors noticed that 20 to 25 design variables are required to fully cover the design space. De Gaspari and Ricci [21
] developed a framework to design camber morphing solutions which consists of two steps: (i) first a gradient-based optimization of the aerodynamic shape of the airfoil is done for a given objective; and (ii) then an internal truss-like structure is optimized such that is able to achieve the desired morphing shape, using a genetic algorithm. Antunes and Azevedo conducted their ASO studies using genetic algorithms [22
]. Surrogate models based on machine learning have also been studied for ASO of airfoils using both gradient-free [23
] and gradient-based [25
] algorithms. With these surrogate models the computational burden of ASO can be relieved at a cost of generating large databases. The computational cost of the latter can overpass the former if only a reduced number of optimization iterations are needed as noted by Bouhlel et al. [25
Despite these optimization algorithms having been employed in several wing and airfoil applications, their combined application is much less common in the literature. ASO is normally done before the TO as in [26
], although there are some papers where two optimizations process are carried out simultaneous for performance [27
] or aeroelastic applications [28
]. Maute and Reich [26
] developed a sequential framework to design morphing structures based firstly on ASO and secondly on TO. James et al. [27
] coupled both optimization processes in their Multidisciplinary Design Optimization (MDO) framework and compared it with the sequential approach for the design of the Common Research Model (CRM) wing. They noticed that their coupled approach was able to achieve a wing model with 18% less drag than with the sequential approach. Gomes and Palacios [28
] applied a fully-coupled aerodynamic and stiffness-based TO methodology for designing a compliant airfoil using FSI. Their methodology consists in two sequential steps: (i) firstly, the airfoil shape is optimized for a given aerodynamic-related objective and a set of constraints considering both ASO and TO; (ii) followed by an inverse design where the internal structure is optimized for a mass minimization objective.
These works [26
] use the Solid Isotropic Material with Penalization (SIMP) model [29
] to parameterize the structural domain, while in the present work the Bidirectional Evolutionary Structural Optimization (BESO) model [30
] is used. BESO has the main advantage of being able to both remove and add material, and it has been applied to solve both academic and industrial problems [31
]. This model has been applied to the structural design of a hypersonic wing considering aerothermoelasticity by Munk et al. [32
]. For the present work, the authors developed a sequential framework to optimize airfoils considering both ASO and TO. To the authors’ best knowledge TO with BESO has not been coupled with ASO for airfoil applications.
This paper is organized as follows: first a literature review is provided in Section 1
; followed by the theoretical background required to conduct aerodynamic shape optimization and topology optimization of airfoils in Section 2
; then the methodology employed in this work is presented in Section 3
; followed by the obtained results in Section 4
; and ending with the concluding remarks drawn from this study in Section 5
5. Concluding Remarks
This research aimed at coupling both aerodynamic and structural designs of airfoils using efficient optimization tools such as aerodynamic shape optimization and topology optimization. Firstly, based on the analysis carried out along the process to reach the stated target, it allowed the researchers to conclude that this field has the capacity to develop highly-influential new designs.
In terms of the aerodynamics, the study has shown that, for very small geometric modifications, noticeable changes in the behavior were found. For instance, the shock-wave was softened even for small shape changes of , leading to considerable drag reductions of 13.9% and 29.04% for case 1 and 2, respectively. Then, the research showed that by improving the aerodynamic behavior of the airfoil, the structure stress state is alleviated.
Discussing then the inner topology distribution, the BESO method has proved to be a robust and practical optimization algorithm since it allows for adding material besides removing it from the structural domain.
The development of a sequential methodology for both optimization processes has enabled the aerostructural optimization of airfoils. This approach has shown synergies between both optimization processes since by designing the airfoil to reduce drag for a given lift coefficient, the load is also reduced and as a consequence the material required to withstand the aerodynamic load is also decreased.