Aerodynamic Design Optimization: Challenges and Perspectives

Antony Jameson pioneered CFD-based aerodynamic design optimization in the late 1980s. In addition to developing the fundamental theory, Jameson implemented that theory in codes that were practical enough to be used in industry. As a result of Jameson’s seminal eﬀorts, a research community has been established in aerodynamic design optimization. This research area has experienced sustained improvements in CFD solvers, mesh deformation, sensitivity computation, and optimization tools. We review recent developments for each of these components and present open-source tools available for aerodynamic shape optimization. A variety of applications is presented, including the optimization of a supercritical airfoil starting from a circle, a web application that optimizes airfoils within a few seconds, aircraft aerodynamic and aerostruc-tural optimization, and aeropropulsive optimization. We also review the Aerodynamic Design Optimization Discussion Group (ADODG) benchmarks and other aerodynamic shape optimization problems. Among the ADODG benchmarks, we focus on the RANS-based problems and discuss some of the issues encountered, including comparing Euler and RANS results and design-space multimodality. The availability of these benchmarks and the open-source tools is expected to enable further studies and benchmarks in CFD-based aerodynamic design optimization and MDO.


Introduction
The development of computational fluid dynamics (CFD) and advances in parallel computing hardware revolutionized the aerodynamic design process.CFD is now routinely used in industry, and it is hard to imagine a design process without using it.While CFD has improved the design process, there is a potential to improve it even further by integrating it with numerical optimization to perform design optimization.
When CFD was first introduced in industry, it was met with skepticism by seasoned aerodynamicists, who were, in part, concerned with having their role diminished by this new tool.Eventually, it became clear that CFD was just that: a new tool that aerodynamicists could leverage to analyze and understand the design.However, this required aerodynamicists to learn this new tool and to adjust their role in the analysis and design process.
Because the end goal is not just to understand the design, but also to find out how to make the design better, CFD analysis is not enough.Coupling CFD analysis with numerical optimization emerged as a way to improve designs more effectively.However, CFD-based aerodynamic design optimization faces a predicament similar to CFD when it was first introduced to industry decades ago.This new tool is set to change the aerodynamic design process, but it faces several challenges.Although the new generation of aerodynamicists tends to have a strong CFD background, a stronger background in design optimization is increasingly important.
In this paper, we focus on significant challenges that must be addressed to broaden the use of aerodynamic design optimization in industry, and how these challenges are being addressed.These challenges include: 1) CFD solver robustness, 2) scalability with number of design variables, 3) efficient and accurate gradient computation, 4) geometry parametrization, 5) robust mesh deformation, 6) availability of specialized software, 7) practical industrial constraints, and 8) consideration of aircraft design disciplines other than aerodynamics.
This paper reviews the efforts to address these challenges.Challenges 1 through 7 pertain more specifically to aerodynamic shape optimization, while Challenge 8 broadens the scope to aircraft design optimization.Some of these challenges were cited in the CFD Vision 2030 study [1].One of the "grand challenges" outlined in that study is the multidisciplinary analysis and optimization of a highly flexible advanced aircraft configuration, which relates directly to Challenge 7, but requires all the other challenges we list to be addressed as well.Many of the developments described in the present paper contribute directly to milestones listed in the CFD Vision 2030 study.
The Aerodynamic Design Optimization Discussion Group (ADODG) benchmarks played a crucial role in the latest developments.Developing optimization problems that everyone could run and compare has motivated researchers to do a specially thorough job of getting the best possible optimized shapes and addressing computational bottlenecks.It also motivated us to make all the results publicly available and to generate benchmarks within and beyond the scope of the ADODG.Finally, solving and discussing the ADODG benchmarks helped us understand our results and their limitations, motivating further developments.
The remainder of this paper is structured as follows.We explain the significance of Antony Jameson's groundbreaking contributions in Sec. 2, together with the developments in aerodynamic design optimization that have ensued.In Sec. 3, we present an overview of the developments that address the challenges listed above.In Sec. 4, we summarize the ADODG benchmarks and cite the efforts towards solving them.In Sec. 5, we highlight applications that we developed and investigated beyond the ADODG benchmarks.We end with concluding remarks in Sec.6.

Jameson's Contributions
The single most important development in aerodynamic shape optimization was the adjoint method, which computes derivatives of performance metrics of interest with respect to design variables.Together with gradient-based optimization, the adjoint method made it possible to optimize shapes parameterized using many variables.This was a crucial development because wing shape requires hundreds of design variables to fully utilize design optimization.
The adjoint method was first developed to solve optimal control problems, where it was used to optimize rockets and aircraft trajectories that were soon after that validated in flight tests [2].An adjoint method was also developed to compute derivatives of finite-element models for structural optimization [3].Pironneau [4] extended the adjoint method to achieve minimum drag shapes for Stokes flow and for the incompressible Euler equations [5].
Antony Jameson then made a sequence of significant breakthroughs that made adjoint-based aerodynamic design optimization practical enough for industrial use [6].He derived the adjoint for the compressible Euler equations and applied it to airfoil design [7] and wing design [8].In this early work, the "control theory" was often cited because the trajectory in optimal control became the shape in aerodynamic optimization [9].
Hicks and Henne [10] had already integrated CFD with numerical optimization and demonstrated the potential for aerodynamic design optimization, developing what became known as Hicks-Henne functions to apply shape perturbations to parametrize the airfoil.However, they were limited to 11 design variables because they used finite-differences to compute the required derivatives.On the other hand, Jameson [11] optimized shapes by changing the positions of thousands of surface mesh points, thanks to the efficiency of the adjoint method.He avoided issues with high-frequency shape oscillations by using a gradient-smoothing technique.Reuther and Jameson [12] implemented B-splines and Hick-Henne functions in their adjoint-based framework and compared the resulting shapes.
Reuther et al. [13] extended the applicability of adjoint optimization to large-scale problems with fullconfiguration, multiblock CFD meshes through improvements to the design parametrization method, mesh perturbation scheme, and parallel implementation.This culminated in the optimization of full configurations considering multiple flight conditions [14,15].The development of the RANS adjoint [16][17][18] enabled the realization of more practical designs, especially in the transonic regime.[19] details the approach and sequence of developments.Since these early efforts spearheaded by Antony Jameson, aerodynamic design optimization has benefited from many developments, in parallel with improvements in the CFD algorithms and computing hardware [6].
We review some of these developments in the remainder of this paper.

CFD solver robustness
Integrating CFD in a numerical optimization cycle demands additional requirements on the robustness of the CFD solver.One of the reasons for this is that a CFD solver is more likely to fail during optimization because the optimizer does not have the intuition that a designer has, and therefore the optimization algorithm tends to provide bad design shapes to the CFD solver at some point during the optimization.Therefore, it is crucial that the CFD solver be able to solve for designs that might not make much sense.
Another reason why robustness is important is that in multidisciplinary analysis, the CFD solver is coupled to other disciplines that might provide unusual boundary conditions for the flow [1].
If the CFD solver fails to converge during an optimization iteration, it interrupts the optimization process, which must then be restarted.Some optimizers allow the user-provided functions to return a "fail" flag, which the optimizer accounts for by taking a less ambitious step for the next function evaluation.While this might prevent the optimization from exiting outright, the optimization might still fail nevertheless because the information from failed function evaluations is not as rich as a normal evaluation.Inaccurate information is still better than a "fail", as long as it is sufficiently accurate to provide the correct trends-having the right sign in each gradient component is usually sufficient to guide the optimizer away from a bad design.Therefore, it is usually worthwhile to force the CFD solver to converge to a solution, even if that solution does not represent the real physics.
To this end, we developed a Jacobian-free approximate Newton-Krylov strategy for robustly solving the RANS equations for a wide range of geometries.Yildirim et al. [20] describes this strategy in detail, and we only include a brief summary here.Figure 1 shows an overview of the method.The solver updates the state at each nonlinear iteration using the backward Euler time stepping formula.To alleviate the difficult startup phase, it uses an adaptive CFL algorithm based on pseudo-transient continuation [21].This starts with a small time step for robustness, and then increases the step size rapidly as it approaches the solution.
This exploits the favorable stability properties of the backwards Euler method during the initial stage, while approaching a Newton-type algorithm as the time step approaches infinity.At each nonlinear iteration, the update vector is obtained by inexactly solving a large linear system.We use two levels of approximation for the exact flow Jacobian residuals (R 0 ): R 1 and R 2 .R 1 is an approximation that leaves out several terms, but it is still closer to the exact Jacobian than R 2 (which is a first order Jacobian approximation).R 1 is used for the GMRES matrix vector products and R 2 is used for the preconditioner.Since this is inexact, we make sure that the thermodynamic quantities do not vary by more than 20%, and we use a line search with backtracking to insure a decrease in the unsteady residual norm.
We implemented the ANK approach in the ADflow open-source CFD solver. 1 ADflow can converge steady state solutions to the RANS equations even if the flow field is inherently unsteady.This is owing to the backwards Euler algorithm, which can stabilize the physically unstable modes.An example of such a flow solution is shown in Fig. 2, where NASA's Common Research Model (CRM) is solved for 90 deg angle of attack at M = 0.85.The solution in this case is not physical, but the important thing is that the flow solver is able to converge.This case was inspired by the 90 deg angle-of-attack airfoil solution by Burgess and Glasby [22].
An example of such an optimization is the one performed by He et al. [23] using MACH-Aero, which starts from a circle and converges to a supercritical airfoil shape in a single fully automated optimization, as shown in Fig. 3. Again, the flow solution for the circle and some of the intermediate shapes is not physical, but the derivatives with respect to shape have the right trend for drag reduction, and as the shape approaches the optimum, the RANS solution becomes valid.As long as the optimizer can eventually converge to a case where the solution is valid, inaccurate intermediate results are irrelevant.

Scaling with number of design variables
Aerodynamic shape optimization requires a large number of shape design variables to achieve the best possible performance.For transonic wing design optimization, Lyu et al. [24] showed that at least 200 shape variables are required to take full advantage of aerodynamic shape optimization, and that beyond this number of variables, there are diminishing returns.Only gradient-based optimization algorithms can handle this number of variables efficiently.Figure 4 shows the scalability of two gradient-free optimization algorithms (NSGA2 and ALPSO) compared to two gradient-based optimization algorithms (SLSQP and SNOPT) for a multidimensional Rosenbrock function.The results in this figure demonstrate that gradientbased algorithms are the only viable option for more than 100 design variables.Furthermore, for a given gradient-based optimizer, there is a large difference between using finite differences (the default way gradientbased optimizers compute the gradients) and an analytic method (which in this case is based on the symbolic differentiation of the function and is thus fast).Therefore, there is a big motivation for developing methods for computing gradients efficiently.We discuss such methods in the next section.
To compare gradient-based and gradient-free optimization algorithms for a more practical problem, Lyu et al. [25] benchmarked various optimizers for a CFD-based wing twist optimization problem.The problem was limited to nine twist design variables and a rather coarse mesh so that the optimization with the gradientfree algorithms was achievable.The optimization problem was to minimize the drag coefficient subject to a lift coefficient constraint.Most gradient-based algorithms achieved optimality within 14 to 230 function evaluations, while the gradient-free algorithms required over 8,000 evaluations.Pulliam et al. [26] found similar trends for an airfoil optimization problem.
For the vast majority of our work, we use SNOPT, a gradient-based algorithm that implements sequential quadratic programming and can handle nonlinear constraints [27].To facilitate its use in our framework, we have wrapped SNOPT (a Fortran library) with Python through the pyOptSparse wrapper [28]. 2 In addition to facilitating the integration with the other modules of the framework, pyOptSparse provides a common interface to various optimization algorithms; to try a different algorithm, only a flag needs to be changed in the main run script.Using this feature, we were able to run all the results shown in Fig. 5 with minimal setup effort.

Effective computation of derivatives
As illustrated in Fig. 4, a good gradient-based algorithm is not sufficient; it is also necessary for the gradients to be computed accurately and efficiently.As mentioned earlier, the adjoint method pioneered by Jameson enables the accurate and efficient computation of gradients in high-dimensional spaces, which addresses this need.While adjoint methods are highly desirable for their efficiency, they also require a large implementation effort.Thus, by effective we mean that the approach for adjoint development provides a good balance between implementation effort, accuracy, and efficiency.
The early adjoint method developed by Jameson used the continuous approach [9].Using this approach, he used calculus of variations on the partial differential equations (PDE) that govern the flow to obtain the adjoint equations and then discretized these equations to solve them for the desired derivatives.
In contrast, the discrete adjoint approach starts with the discretized governing equations and differentiates those equations to obtain a linear system that is then solved to obtain the desired derivatives.When comparing the continuous and discrete approaches, Nadarajah and Jameson [29] concluded that the discrete approach was more costly than the continuous approach and that there was no benefit in the discrete approach.However, the discrete approach has been the preferred approach in the last decade because it is more straightforward to develop and because its gradients are consistent with the discretization [30][31][32].
To address the issue of the large implementation effort, we have developed a general recipe for effective adjoint method implementation that uses automatic differentiation (AD) to derive the terms needed for the discrete adjoint [33].This approach had been previously mentioned by Giles and Pierce [30] and recommended in the survey by [32].We have applied this approach to implement discrete adjoint methods to both ADflow [33] and OpenFOAM [34,35].Similar approaches have been followed in the most recent adjoint implementations of the SU2 [36] and STAMPS flow solvers [37].
One unique feature of our approach is that we adopt a Jacobian-free GMRES strategy for the solution of the adjoint equations, which we found to be the most scalable [33].The overall approach is illustrated in Fig. 6 using an extended design structure matrix (XDSM)representation [38].
Figure 6: XDSM of the hybrid-adjoint approach [33].The transpose of the Jacobian [∂R/∂w] T P C is computed using forward mode AD and coloring in the 0, 2→1 loop.The adjoint equation right-hand side vector [∂f /∂w] T is computed using reversemode AD in step 3.Then, the adjoint equation is solved using GMRES and the resulting adjoint vector is used in step 8 to compute the desired total derivatives.

Geometry Parametrization
The early adjoint-based aerodynamic design efforts optimized shapes by directly changing the positions of all the CFD surface mesh points.However, it is desirable to separate the geometry representation from the flow solver discretization to isolate the effects of mesh refinement from those of geometry parametrization refinement and approach.This separation is also important for modularity in computational framework settings, especially when performing multidisciplinary design optimization (MDO).
To parameterize geometry for optimization, computer aided design (CAD) is the natural choice given its universal use in industry, and because it allows designers to parametrize the geometry using variables that are intuitive to them.This was also identified as a need in the CFD Vision 2030 report [1].Researchers have attempted to integrate CAD in aerodynamic shape optimization, but the computational cost and accuracy of the derivatives of the CAD largely negate the advantages provided by the adjoint method [39,40].
Instead, most researchers have implemented their own CAD-free geometry parametrization approaches, such as free-form deformation [41] and B-splines [42].There have been various studies comparing the different approaches, concluding that none of the geometry parametrization methods is superior to the other overall [43][44][45][46][47].

Mesh deformation
A shape optimization cycle requires a geometry engine to translate new shape design variables to new shapes and automatically obtain a new mesh for the new design surface.The new mesh is not re-generated, but rather, it is a deformation of the baseline mesh.The robustness of the mesh deformation is essential for the same reasons cited for flow solver robustness in Sec.3.1: If it fails, it jeopardizes the optimization process.
To address this need, we implemented IDWarp, an efficient analytic method for volume mesh deformation3 , which was also crucial in the airfoil optimization starting from a circle described in Sec.3.1 [23].
The implementation is based on the inverse distance weighting proposed by Luke et al. [48], with some improvements.The most significant of these improvements is the efficient computation of derivatives via reverse-mode AD.More specifically, IDWarp computes the derivatives of all mesh point coordinates with respect to changes in the surface mesh point coordinates.This is one of the partial derivatives in the derivative chain between the aerodynamic force coefficients and the shape variables.Figure 7 shows examples of mesh deformations performed with IDWarp for a wing mesh.

Software availability
One of the reasons why aerodynamic design optimization has not been more widely used is that it requires a complex set of tools that have not been readily available in the aerodynamic design community at large.Commercial CFD solvers have not implemented adjoint solvers until recently, and they have not been developed with optimization in mind.Other CFD-based optimization capability has been restricted to a few research groups in academia and government laboratories.One exception is the open-source CFD solver SU2, which includes an adjoint solver and provides aerodynamic shape optimization capability [49].
The capabilities described in the present paper are integrated in the MACH-Aero framework, which is available under an open source license. 4An XDSM showing the aerodynamic shape optimization in MACH-Aero is shown in Fig. 8.All the components shown in the diagonal are wrapped in Python for modularity and ease of use.The optimizers are provided through the pyOptSparse interface introduced in Sec.3.2.The geometry parametrization can use either our free-form deformation (FFD) implementation, pyGeo [41], or OpenVSP [50].The volume mesh deformation is provided by IDWarp, which we introduced in Sec.3.5.
Either ADflow or OpenFOAM are available to solve the flow using a common Python interface.We envision that other CFD solvers could be easily integrated by wrapping them using the same Python interface.Finally, as previously mentioned, both CFD solvers include and adjoint solver to efficiently compute the gradients for design optimization.Even though gradient-based optimization with the adjoint method enables efficient aerodynamic shape optimization, it still requires hours in a parallel computer to fully converge an optimization.To make aerodynamic shape optimization more accessible, we have developed a web-based data-driven approach to airfoil design that takes just a few seconds for optimization [51]. 5Figure 9 shows a screen shot of this online tool, which includes a database of over 1,500 airfoils.

Practical industrial design constraints
Based on our collaborations with industry, we identified several constraints that had to be enforced, which required new developments.Geometric constraints, such as variable fuel volume, wing thickness, leading edge radius, and trailing edge angle constraints are linear and were relatively easy to implement [24].
To consider geometric constraints implicitly, it is also possible to use a data-driven approach [52].Other constraints, such as buffet and flutter, are highly nonlinear and require much more development effort.Buffet is a phenomenon that is caused by shock-induced separation and is undesirable because it causes vibration.We developed a constraint formulation for buffet based on a separation sensor function [53], which was shown to be effective in various applications [54][55][56][57][58][59].We have found that constraining buffet is crucial in transonic wing design.In Fig. 10 we compare the fuel burn contours for a single-point optimization without buffet constraints to a multipoint optimization with buffet constraints.The unconstrained singlepoint optimization achieves a high performance outside the buffet margin and thus this high performance is not usable in practice.The buffet-constrained optimization moves the area of high performance and the 30% margin boundary so that the highest performance region is usable.
We developed an approach that is mathematically identical to constrain cavitation in hydrofoil design optimization [60,61].This approach could also be potentially used for low-speed and high-lift performance, where we would formulate a minimum lift coefficient constraint to be achieved without major separation.
We also made progress towards constraining flutter using two different approaches [62,63].Jonsson et al. [64] review existing methods for constraining flutter in optimization, including the approaches cited above.
Like flutter, many other practical constraints require the consideration of disciplines beyond aerodynamics, which is the subject of the next section.

CFD-based multidisciplinary design optimization
Aerodynamics is not enough to achieve a feasible aircraft with high performance.One of the most important other disciplines is structures, which couples with aerodynamics to determine the wing performance.The coupling between the disciplines manifests itself both in analysis and design.The analysis requires solving a coupled system with both disciplines because the aerodynamic model provides the forces to the structural model, while the structural model determines the displacements and hence the aerodynamic shape.
From the design point of view, there is a trade between structural weight and aerodynamic drag that depends on the chosen objective function.Objective functions such a fuel burn and take-off weight depend on both structural weight and drag, but in different proportions.Fuel burn places more emphasis on reducing drag, while take-off weight places more emphasis on reducing structural weight.More broadly, multidisciplinary design optimization (MDO) provides a way to model a coupled system and find optimal multidisciplinary design trades [65].
Given the motivation for efficient and accurate gradient computation, we developed a coupled-adjoint approach that computes gradients accounting for the aerostructural coupling [66].To achieve this, we coupled ADflow to the open-source structural solver TACS, which has an adjoint solver [67].The result was the MDO for aircraft configurations with high fidelity (MACH) framework.Using this approach, we can compute derivatives of aerodynamic forces and structural stresses with respect to both structural sizing and aerodynamic shape variables.MACH enabled us to perform the simultaneous design optimization of aerodynamic shape including wing planform variables and structural sizing for various aircraft design applications, including planform optimization starting from the CRM baseline [57], multimission fuel burn minimization [68], tow-steered composite optimization [59], and morphing wing optimization [69,70].The MACH framework and these applications directly address the CFD 2030 Vision technology roadmap, which calls for standard for coupling to other disciplines and CFD-based optimization of the entire aircraft [1].
The coupled-adjoint approach has also been generalized for an arbitrary number of disciplines [71], which lead to the modular analysis and unified derivatives (MAUD) architecture for MDO [72].The MAUD architecture has been integrated into and improved upon in the OpenMDAO framework [73].
OpenMDAO has facilitated the coupling of CFD with other disciplines and the corresponding coupled adjoint derivative computations.Other MDO work beyond aerostructural optimization includes the simultaneous optimization of aerodynamic shape and propulsion system [74], and optimization of wing, mission, and allocation [75,76].

ADODG benchmarks
The development of the ADODG benchmarks was motivated by the fact that no such benchmarks were in place when the ADODG was formed.Each researcher had solved different problems using different approaches, making it difficult to compare the various approaches.The ADODG benchmarks enable researchers to solve the same problem so that the results are more comparable.A series of aerodynamic shape optimization problems of increasing complexity in the geometry and physics was developed.
Before the ADODG benchmark cases were established, the MDO Lab had been focusing on CFD-based wing aerostructural design optimization, which we introduced in Sec.3.8.However, our early efforts solved the Euler equations for the aerodynamics and it was difficult to interpret the optimal design trends because they included both aerodynamic and structural effects.The ADODG benchmark cases (and Case 4 in particular), forced us to do more detailed studies considering aerodynamics alone.Understanding the aerodynamic shape optimization results then helped us understand the aerostructural results a lot better.

Summary of the ADODG benchmark cases
The ADODG benchmark problems are listed in Table 1, together with the references that tackled the respective problems.The objective for all problems is to minimize the drag coefficient (or a sum of drag coefficients in the multipoint cases).Most cases are subject to a lift coefficient constraint as well as thickness constraints.Some cases include a moment constraint as well.
The two first cases are airfoil shape optimization problems: Case 1 starts from a NACA 0012 airfoil baseline and and Case 2 starts from the RAE 2822 airfoil.These two cases use different aerodynamic models: Case 1 solves the Euler equations, while Case 2 is based on the RANS equations.Another difference between these two cases is that Case 1 does not enforce a lift constraint, while Case 2 does.Based on the number of references, these two cases have been the most solved by far.This is in large part owing to the lower cost of solving two-dimensional cases.
Cases 3 and 4 optimize wing shapes; Case 3 is a subsonic case that solves the Euler equations, and Case 4 is a transonic case using RANS.Case 3 optimizes only twist, while Case 4 includes both twist and airfoil shapes.All wing cases include a volume constraint, where the volume of the optimized wing cannot decrease relative to the baseline wing.Finally, Case 6 is a wing shape optimization problem similar to Case 3 with the addition of airfoil shape variables and planform variables (chord variation, sweep, span, and dihedral).The idea is to provide as much freedom in the design space exploration as possible and to determine the characteristics of the wing design problem in terms of multimodality.

Euler-versus RANS-based optimization
Although Cases 1 and 3 do not represent the real physics, they challenge the numerical methods of the CFD, the parametrization approach, and the optimizer.Case 1 in particular has received much attention even though the resulting shape is of no practical value.Such cases do have some value and Case 1 in particular has inspired and demonstrated new techniques.However, beyond a certain point, one might fall into the trap of developing solutions to issues that do not exist in practice.While RANS is more costly, many of the issues encountered with Euler-based shape optimization, such as non-unique solutions, disappear when using RANS.In addition, we have found that for transonic wing shape optimization, Euler models resulted in completely different shapes, as shown in Fig. 11 [100].The RANS-optimized pressure distribution is better behaved and the optimal airfoils show large differences in shape [100].

RANS-based wing design optimization
As soon as the ADODG benchmark cases were established, we focused our first efforts on Case 4. This was because this was the most similar to the wing aerostructural design optimization that we had already been performing (see Sec. 3.8) [101].
Lyu et al. [24] reported our first results on Case 4.1, which was a single point optimization.The resulting wing, shown in Fig. 12, exhibits much thinner wing sections than the baseline CRM in the outboard sections, which also have sharper leading edges.This is because the case specifies that the airfoil thickness is allowed to be as low as 25% of the original CRM wing.The optimizer exploited this generous allowance to reduce the viscous pressure drag on the outboard of the wing, while thickening the inboard of the wing to satisfy the volume constraint.The thickening of the inboard increases the viscous pressure drag, but the decrease in the outboard more than compensates for this increase.Since the chord in the inboard is much larger, it is more efficient to increase the volume in this region to satisfy the volume constraint.Méheut et al. [94] verified this explanation by analyzing our optimal geometry with a drag decomposition tool.Not everyone achieved the same result.In some cases the problem was not setup exactly as stated, but we suspect that in other cases, the gradients were not accurate enough.We found that the trade between outboard and inboard viscous pressure drag is subtle and requires a large number of iterations with accurate gradients for the optimizer to find it.
Lyu et al. [24] also included the solution for a five-point multipoint problem before the ADODG added a series of multipoint problems to Case 4. The optimized shape for the multipoint case has less sharp leading edges.While the multipoint optimum does not exhibit a shock-free solution like its single-point counterpart, it represents an optimal compromise between the different flight conditions with a much more robust solution over the flight condition space.Various multipoint cases (Cases 4.2-4.7)where then added to the Case 4 ADODG benchmark, ranging from three to nine points.Kenway and Martins [102] solved and discussed all these cases, including a postoptimality study of the performance of the optimal wings over a range of flight conditions (defined by the lift coefficient and Mach number).The corresponding contour plots are shown in Fig. 13, where the contours were obtained by evaluating M L/D in a grid of RANS solutions for each optimal wing.In these plots, we can see that the larger the number of points considered, the more robust the design is to variations in the flight conditions.This robustness is achieved with a small penalty in the maximum performance.One particularly interesting result is that of Case 4.5, which results in two maximum performance points that have region of lower performance between them.

Multimodality in aerodynamic shape optimization
A popular belief in the research community is that aerodynamic shape optimization problems are often multimodal.However, our studies have found that this concern is largely unwarranted.It is impossible to prove that we have found the global optimum, but to show that the problem is multimodal, we only need to find a second local minimum.Therefore, as long as the design space is explored adequately, it is fair to assume that an optimum that has been obtained consistently is the global one until proven otherwise.
As we describe in this section, we have shown convergence to unique optimal consistency for most of the ADODG cases, adding evidence to the hypothesis that these problems are unimodal and that we have found the global optimum.
In our first study of Case 4, we tried to find multiple local minima by starting from random perturbations of the original shape [24].All optimizations converged to essentially the same shape.While we saw some difference in the shape design variables, the differences in drag between the various designs was only 0.1 drag counts.Follow-up work by Yu et al. [96] found that refining the computational mesh brought these optimized shapes even closer to each other.That work included a starting design that consisted of a CRM planform with NACA 0012 airfoils and no twist, which also converged to the same Case 4 optimum.He et al. [23] did a similar multi-start study for Case 2 and found that the optimization always converged to the same optimum airfoil.Therefore, our hypothesis is that the design space for twist and airfoil shape optimization is unimodal from the physical point of view.Around the global optimum, however, numerical noise causes gradient-based optimizers to converge to slightly different designs.
We believe that researchers often identify spurious multiple local minima because of numerical issues with either the computed gradients or the optimization algorithm.A careful verification of the computed gradients against an equally accurate method is recommended.In ADflow, the adjoint gradients have been verified against the complex-step method [103].We also recommend that practitioners pay close attention to the optimality and feasibility tolerances, as well as convergence histories.When using gradient-free algorithms, claimed optima are even more suspect because unlike the mathematical optimality criteria of gradient-based algorithms, the optimality criteria for most gradient-free algorithms are based on heuristics.

Full configuration aerodynamic optimization and MDO
Case 4 considers only the wing from the Common Research Model (CRM) configuration and compensates by the lack of horizontal tail by imposing a moment coefficient constraint.While this design freedom provided an excellent test for the overall optimization procedure by requiring robustness to changes in the shape and accurate gradients, the optimal wings were too thin to be practical.
Case 5, builds upon Case 4, by considering the full CRM configuration (wing, fuselage, and horizontal tail), also known as the Drag Prediction Workshop (DPW) 4 geometry [104].By adding back the fuselage and horizontal tail, we can enforce trim by changing the horizontal tail rotation, and capture the true effect of the wing shape on trim drag.Chen et al. [97] investigated this effect and created a surrogate model for the trim drag to use when the horizontal tail is not available in the CFD model.
As mentioned in Sec.3.8, aerodynamics is not enough for aircraft design.Inspired by the success of the ADODG benchmarks, we developed an aerostructural wing design benchmark based on Case 5.This required creating a wingbox structure (loosely based on the Boeing 777 structural arrangement) and a new jig outer mold line [57].The structure and shape were designed such that they achieve the shape of the original CRM when analyzed using static aeroelastic analysis at the CRM nominal flight condition (C L = 0.5, M = 0.85).
The benchmark is called the undeflected CRM (uCRM) and has been used in various studies [59,69,70,105].

Other applications
The CFD-based optimization efforts are summarized in Table 2.They include the applications that were already mentioned as well as two other applications that go beyond aircraft applications: the aerodynamic shape optimization of wind turbine blades and the hydrodynamic and hydrostructural optimization of hydrofoils.
While we are not involved in experimental work, we have had two optimization results that have been the subject of experimental measurements.One was the hydrofoil design optimization of Garg et al. [60], who was built and tested in a water tunnel.The experimental results matched the numerical predictions [61].The other result that was tested was the tow-steered aerostructural optimization of Brooks et al. [59].Aurora Flight Sciences built a one-third-scale model of the optimized wingbox [106] that underwent structural testing at NASA Armstrong Research Center. 6

Conclusions
In this paper, we honor Antony Jameson's legacy by recalling his seminal contributions in the development of the adjoint method.We then review the significant developments of the last decade towards enabling CFDbased aircraft design optimization.The cited references provide much more detail on the methods and their applications.The common denominator in all of this work is accurate and efficient computation of derivatives via adjoint methods, which, together with gradient-based optimization, enables the solution of large-scale CFD-based aerodynamic shape and MDO for aircraft configurations and other applications.Over the last several years, several challenges were tackled to make the optimization more scalable, more robust, and more practical.Much of the developed software, including all the components required for aerodynamic shape optimization, are available under an open-source license. 7he ADODG benchmark cases motivated many of the reported developments.Even cases that are not realistic from the practical point tested the methods' limits, which ultimately motivated improvements in accuracy and robustness.Exceptions to this are the Euler-based cases, which include issues that we do not think are worth tackling because they disappear when using RANS.Working closely with industry has been invaluable in identifying the challenges that needed to be solved for practical applications.
Given the contributions above and the fact that much of the code that we developed is available under an open-source license, we expect that the use of CFD-based optimization will increase and become more widespread than ever.The recent developments in the OpenMDAO framework, including coupled-adjoint derivative computation, have facilitated the implementation of CFD-based MDO, and we expect further developments in this area.
Jameson's research approach has been remarkable because he was not satisfied with just developing theory; he also put a tremendous effort into implementing his methods (by himself!) and testing those implementations in real-world applications in direct collaborations with industry.This research approach provides an exemplary model for other researchers to follow that has inspired the author and his research group.

Figure 4 :
Figure 4: Gradient-based optimization is the only hope for handling the large numbers of design variables required for aerodynamic shape optimization [25].

Figure 5 :
Figure 5: Optimizer comparison for a wing design problem with nine twist variables shows that gradient-free methods cost 2-3 orders of magnitude more to optimize this simple problem [25].

Figure 7 :
Figure 7: Mesh deformations for wing planform optimization showing twist, sweep, and span variables (courtesy of Ping He).

Figure 9 :
Figure 9: Webfoil is an online database and airfoil optimization tool.

Figure 10 :
Figure 10: Fuel burn contours and buffet boundary over the space of flight conditions (lift coefficient and Mach number) for a single-point optimization without buffet constraints (left) and multipoint optimization with buffet constraints (right).The red line represents the buffet boundary and the blue line represents the 30% margin to the boundary, which much not be exceeded in cruise flight conditions [53].
drag minimization of the CRM wing-body-tail configuration at flight Reynolds number Case 5 is an extension of Case 4 that adds the fuselage and tail.The baseline shape for Case 4 is actually the wing from the CRM full aircraft configuration clipped at the fuselage intersection.Case 5 restores the full CRM configuration and provides a benchmark for a more realistic and industrially relevant design problem.It uses the flight Reynolds number (as opposed to the wind tunnel one) and the thickness constraints prevent the thickness from decreasing relative to the baseline CRM wing.It also includes a tail rotation angle design variable to trim the aircraft at various flight conditions.Case 5 adds two off-design flight conditions to prevent buffet [53].

Figure 11 :
Figure 11: Comparison between Euler-and RANS-based aerodynamic shape optimizations starting from the ONERA M6 wing.

Figure 13 :
Figure 13: Aerodynamic performance in the flight condition space for all optimal wings for Case 4 [102].

Table 2 :
List of optimization problems solved with MACH.