1.1. Background
The aerodynamic behavior of a wing is tightly coupled to its structural response, and vice versa. Before the advent of modern multidisciplinary analysis, predicting the aeroelastic response for a given wing was limited. This limitation led to the development of the traditional wing design process, in which successive designs are passed iteratively between aerodynamics and structures engineering groups. In his seminal book on aircraft design, released posthumously in 1978, Kuchemann [
1] writes:
This should be one of the aims for the future: we want an integrated aerodynamic and structural analysis of the dynamics of the flying vehicle as one deformable body, and to use that for design purposes.
The pursuit of this ideal gave birth to the field of multidisciplinary analysis and optimization (MDAO), wherein integrated aerostructural analysis and design framework is now a reality. Initially, simplified models were used out of necessity because of computational constraints. Haftka [
2] combined a lifting line model with a simple finite-element model to perform one of the earliest aerostructural optimizations. Low-fidelity models continue to be used to facilitate analysis and optimization. Chittick and Martins [
3] used a panel method and a single tubular spar to demonstrate aerostructural optimization. Jansen et al. [
4] used an aerodynamic panel method and an equivalent beam structural model to enable exploration of the nonplanar wing design space using a gradient-free optimizer (which would require too many function evaluations to use with higher-fidelity models). More recently, Jasa et al. [
5] developed an open-source aerostructural model (OpenAeroStruct) in OpenMDAO [
6] that uses a vortex-lattice method for aerodynamics and a beam finite-element model for the structure.
The objective of multidisciplinary design optimization (MDO) is to optimize the design parameters of multiple disciplines simultaneously, rather than sequentially. For MDO to fully replace the traditional wing design process, it must be capable of handling hundreds of design variables. Additionally, high-fidelity computational models are necessary to capture the influence of the design variables on the wing’s performance. For a given optimizer, the number of iterations required to reach a solution increases with the number of design variables. In general, the cost of each optimization iteration increases with the fidelity of the computational analysis being used. Thus, for high-fidelity wing design, it is advantageous to use an optimizer that can handle many design variables while keeping the number of iterations low. Gradient-based optimizers require far fewer iterations to reach a solution than gradient-free methods [
7].
Gradient-based optimizers are faster than gradient-free methods because they use derivatives to determine their path through the design space. However, derivative computations can be expensive. Additionally, the accuracy of the computed derivatives is critical to the success of the optimization. A naïve implementation of derivative computations might use the finite-difference method, which does not generally yield accurate gradients and has a computational cost that is proportional to the number of design variables [
8]. There are more sophisticated approaches to gradient calculation, such as the complex-step approximation [
9] and the adjoint method [
10,
11]. The adjoint method computes derivatives with the same level of accuracy as the primal solver and has a computational cost that is independent of the number of design variables.
The coupled-adjoint for aerostructural systems enables high-fidelity gradient-based optimization of realistic wing designs [
12]. One of the first high-fidelity aerostructural optimizations was conducted by Martins et al. [
13], who optimized the wing shape and wingbox sizing of a supersonic business jet using Euler computational fluid dynamics (CFD) and a finite-element model. Since then, there have been various other efforts using CFD-based aerostructural optimization with both Euler [
14,
15] and Reynolds-averaged Navier–Stokes (RANS) models [
16,
17,
18,
19,
20]. Although more accurate fluid flow models are possible with large-eddy and direct-numerical simulations, the computational cost of such methods renders them prohibitive for wing design optimization with the current technology. Furthermore, RANS is accurate enough for drag minimization at cruise flight conditions [
21,
22]. Thus, RANS coupled with finite-element structural analysis represents the state-of-the-art for aerostructural wing design optimization.
The convergence of a gradient-based optimizer is determined by the Karush–Kuhn–Tucker (KKT) conditions, which ensure that the constraints are satisfied and the objective cannot be locally improved at the final solution. However, despite the rigor of the KKT conditions—or perhaps because of it—gradient-based optimizers are only guaranteed to converge to a single, local minimum. In optimization problems with multiple local minima, a gradient-based optimizer converges to only one solution, which may not be the global optimum. Multiple research efforts have shown that aerodynamic shape optimization (ASO) problems with airfoil shape and wing twist are unimodal [
7,
21,
23,
24]. However, the appearance of spurious, multiple local minima in these types of problems is possible when the convergence criterion of both the functions and derivatives is not sufficiently stringent, as discussed by Yu et al. [
7]. These spurious local minima were also exposed by Koo and Zingg [
24] in a follow-up to a previous paper [
25].
When planform variables are added to the design problem, multiple local minima do appear in the design space [
23,
26,
27,
28]. However, it is crucial to distinguish mathematically rigorous local minima and physically significant local minima. For example, Chernukhin and Zingg [
23] found many local minima in a benchmark design problem with chord, dihedral, sweep, and span variables, in addition to airfoil shape and twist. However, they used Euler CFD in the optimization, thereby creating a nonphysical design space with local minima that might not exist in reality. Streuber and Zingg [
27] and Bons et al. [
26] approached the same benchmark problem using RANS CFD, but still found multiple local minima. However, by carefully studying the influence of each of the design variables, Bons et al. [
26] discovered a physically legitimate reason for multimodality in the chord distribution. By adding a constraint to enforce a monotonically decreasing chord distribution, some multimodality was eliminated from the problem, and more practical designs were obtained. Other common examples of legitimate, multiple local minima include upward and downward winglets, and forward and aft swept wings.
An understanding of multimodality in the wing design space enables designers to pose optimization problems to facilitate design space traversal from the starting design to the global optimum. Many of the optimization problems in the literature involve small changes between the baseline and optimized designs. These refining optimization problems do not demonstrate the optimizer’s ability to traverse the design space, as would be required to find a solution in the design of an unconventional aircraft. Instead, researchers often resort to randomly perturbing the initial design [
21,
27] or starting from a blank slate design [
23,
26]. These design space exploration studies bolster confidence in the suitability of gradient-based optimizers for ASO problems. However, they have not been replicated for aerostructural wing design problems. In the same way that RANS optimization results supersede Euler-optimized designs, the introduction of structures into the design problem creates an entirely new—and more realistic—design space to study. In the current work, we apply similar methods to a more realistic transonic wing design problem with consideration of structures. Adding a wingbox structure allows the optimization to find the proper trade-off between weight and drag as it varies the planform and nonplanarity of the wing.
Single-point optimizations are prone to exhibit poor off-design performance. One of the most common solutions for this problem is to set an objective function that is a weighted average of the performance at multiple design points. The set of design conditions included in the objective is referred to as a
multipoint stencil. Thus, even though the optimization problems solved in
Section 3.2 includes cruise, maneuver, and buffet analysis points, we designate them as single-point designs because the objective function was only based on a single design point. Using a multipoint objective improves the average performance across the stencil at the expense of the nominal design point. However, it can result in intermittent performance, wherein the design functions optimally at the specified design conditions but poorly in the intervals. Drela [
29] reported this phenomenon in a set of airfoil optimization studies and showed that increasing the number of points in the stencil helped curb this tendency. In wing ASO, Lyu et al. [
21] obtained a more robust design using a 5-point stencil than with a single-point optimization. The multipoint design had a weak shock across the stencil, whereas the single-point design had completely eliminated the shock at the nominal design point. They also found that the multipoint design had a larger leading-edge radius than the single-point design. Kenway and Martins [
30] compared different multipoint stencils of varying size and composition and found a good compromise between robustness and computational expense with a carefully chosen 5-point stencil. Various other efforts have performed multipoint ASO successfully [
31,
32,
33,
34,
35]. Multipoint optimization has also been demonstrated in aerostructural wing design [
18,
36,
37].
Although there have been extensive comparisons between single-point and multipoint designs with ASO, the same cannot be said for aerostructural wing optimization. Additionally, most of the past efforts on multipoint design have focused on robust cruise performance without considering the impact of design changes at low-speed, high-lift conditions. Preserving low-speed, high-lift performance in a wing optimization problem is notoriously difficult because of the complications that arise from modeling and parameterizing high-lift devices. The difficulties associated with high-lift devices can be avoided by considering clean wing performance at low-speed, high-lift conditions. To this end, Wakayama and Kroo [
38] and Ning and Kroo [
39] have shown that constraining
using critical section theory results in a more practical planform design. In airfoil optimization, Buckley et al. [
40] added a constraint on
into the multipoint objective function to meet safety requirements at a low-speed condition. Rather than constraining
, Khosravi and Zingg [
14] included climb drag in the multipoint objective function to encourage improvement in that regime. The whole issue is often skirted by simply imposing limitations on the geometric parametrization to prevent changes that would adversely affect high-lift performance (e.g., minimum leading-edge thickness). This work introduces a new approach to preserving low-speed, high-lift performance while minimizing cruise fuel burn.