Optimization in Aerospace
Author names: Daniel Charles
Stewards: Dajun Yue and Fengqi You
Aerospace collectively represents one of the most sophisticated technological endeavors and largest industries in the world. Coming with substantial costs, nearly every aspect of the industry, from aircraft design to material selection to operation, has been optimized in at least one way. Some of the most important examples are explored below.
Wing Shape Optimization
In designing the wing of an airplane, several fluid dynamic concepts come into play. The two most prominent ones are lift and drag, which correspond to a wing's ability to fly the airplane against gravity without wasting energy by moving forward. The behavior of an airfoil, or cross-section, of wings in air is thoroughly studied to make efficient airplanes. Secondly, the overall shape of the wing can be optimized as well. Considerations generally include lift and drag, as with the airfoil, but also noise and stability.
Some of the earliest wing optimization attempts can be traced back to the late 1500s. Newton applied variational calculus to drag minimization problems. The next leap in the field, however, didn't come until the 1960s when numerical solutions became available with advances in hardware. The 70s saw the application of gradient-based models for aerodynamic shapes, solving for up to 11 different variables in a three-dimensional design space. Later methods used the Stokes flow equations with Euluer's incompressible medium equations to solve for sing airfoils (or cross-sections). When iterative computational fluid dynamics (CFD) became feasible for small-mesh models, the solutions became even more robust. 
Even when meeting the numerous technical requirements to make an operable wing, there are several ways of gauging performance.
The lift coefficient, , corresponds to the wings' ability to keep a given weight, , at a horizontal cruising speed , , called the stall speed. For commercial flights, higher lift coefficients of rouhgly 3.0-3.3 are desirable for lifting heavy loads at low speeds during takeoff and ensuring longer periods of safe flight during emergencies. Military aircraft, by contrast, are willing to sacrifice their lift coefficients, going as low as 1.1, in order to have very low drag coefficients. Unlike commercial airplanes, military pilots usually have an eject seat so losing power is non-fatal making maximum speed more valuable than reliable flight.
Another performance consideration is stability. Unlike the lift coefficient, which solely pertains to vertical lift, some wing coefficients are more prone to sudden jerks than others. Again, commercial wings are designed to have significant stability which improves passenger comfort. Military ones, on the other hand, rely on instability to quickly maneuver in the air. Without sophisticated algorithms to keep the plane level, manually flying a modern military jet would be nearly impossible because of its designed tendency to quickly turn or dive at any perturbation.
Depending on the company and program, the most important quality of the wings may be the cost. Some airfoils, like highly cambered or supersonic double wedge sections, are substantially more difficult to manufacture. Therefore, it requires more skilled laborers and may result in a lower production efficiency. In light of that, many aerospace engineers must consider the financial burden of making a particular shape and weigh it against the marginal increase in performance. 
Since there are so many potential objectives when designing a wing, most companies use multi-objective optimization. Therefore, the wing design chosen may not represent the "best" in any single trait, but it likely has the least net trade-offs. A common approach is a simple weighted sum or ratio between the several objectives into some macro-objective, e.g. the lift to drag ratio, . Although the computational requirements are substantial, the largest problem with the method are actually associated with scaling each sub-objective. 
It is not uncommon to find literature that claims to have found a Boeing Dreamliner wing with 8% less drag than the current design. While it may be true, it is exceedingly likely that the authors used a simplified objective function that neglects other important considerations such as manufacturing costs, serviceability, etc.
Design space and variables
An aircraft wing is geometrically complex and can be described by dozens of variables. Some of the most important are described here.
As already mentioned, the airfoil, or cross-section, of a wing plays a critical role in generating lift. Unlike the Wright Brothers who relied on high-area, low-efficiency bi-plane wings, modern optimization tools allow for complicated and variable airfoils to be justified. Since the airfoil is an oriented two-dimensional form, several variables describes its shape. The chord line describes the length from leading edge to trailing edge via the horizontal reference of the airfoil. The optimum angle of attack, AOA, represents the steepness of wing with respect to the horizontal motion direction. In many situations, the AOA is taken as a parameter for the entire wing, although it is possible to calculate an optimum for each airfoil section.
Wings with high sweep angles are becoming more prevalent as they often impart higher fuel efficiency. However, airplane designs are limited by airport gate sizes which constrain perpendicular wing length. The efficiency effect is so tantalizing for airlines that some airplanes in development, like the Boeing 777-X, will have wing extensions that fold away when parked in the gate. A more conventional approach to building a functionally longer wing (while still fitting in the gate) requires sweeping the wing away from the wing axis. Angling the wing forward does this, but comes with unacceptable instability. Therefore, most commercial planes sweep the wing rearwards.
Most wings are also angled up, or down with respect to the horizontal plane of the aircraft. This modification, called a dihedral angle, is used to increase the functional wing length and pitch stability.
The aspect ratio of the wing is closely related to the lifting area, both of which are critical variables in almost any objective function. In the most basic lift equations, the "footprint" of the wing is linearly proportional to the force:
Of course, modern lift calculations involve substantially more information. including altitude and surface topography, in order to estimate the boundary layer thickness.[www.grc.nasa.gov]
Before beginning optimization of a wing shape, several assumptions about the basic structure are made. These typically include the number of wings, vertical position on the fuselage, and sometimes airfoil shape. An initial condition based on lifting line theory is designed by aerospace engineers and used as a starting point for the optimization. Then, CFD is run to gather fundamental performance values (e.g. lift, drag, pitching moment) which are converted into an objective value. The next iteration makes a set of small changes to the geometry and repeats the process until the algorithm converges on a form with the best objective value.
The large number of potential variables requires careful consideration of computational resources and algorithms. While genetic algorithms are a candidate for global optimization of the shape, their dependence on many iterations (each performing CFD) makes it generally infeasible. Therefore, the gradient-based optimizer with adjoint gradient evaluations is chosen to guide consecutive iterations.
After solving with ...
"The optimization algorithm we use for all the results presented herein is SNOPT (sparse nonlinear optimizer)  through the Python interface pyOpt . SNOPT is a gradient-based optimizer that implements a sequential quadratic programming method; it is capable of solving large-scale nonlinear optimization problems with thousands of constraints and design variables. SNOPT uses a smooth augmented Lagrangian merit function, and the Hessian of the Lagrangian is approximated using a limited-memory quasi-Newton method." 
A modern optimization problem utilizes iterative CFD in conjunction with minimization algorithms for a multi-objective model. In essence, a starting design is proposed, CFD performed, results analyzed, shape changed, and repeated.
Navier stokes for airfoil CFD for the wing process (from dwc paper)(from http://ntrs.nasa.gov/archive/nasa/casi.ntrs.nasa.gov/19950018914.pdf)
pages 35-... of DWC
In the above equation, k is the Smeaton coefficient. This describes the drag of a single square foot moving at one mile per hour, and was generally accepted as a constant around 0.005 in the early 1900s. [wright.nasa.gov] In fact, the Smeaton coefficient is a function of velocity and total area as well, which, when accounted for in maximizing lift, adds some fidelity to the model. [www.grc.nasa.gov/]
Because of the limited types of aircraft missions (military and commercial), two main wing designs that dominate the design space with minor differences between aircraft.
Most commercial and military aircraft adopt this wing style, where wings are separate from and extend out of the fuselage. This wing orientation, unlike the blended wing body, typically allows for faster speeds (which improve lift) at the expense of smaller wings (which typically hinder lift).
Blended Wing Body
The blended wing body configuration is highly fuel efficient as it maximizes the lift of the entire airplane while also improving cabin size. Therefore, the commercial implications of this design are drastic. The military has already begun using this platform in the iconic B-2 bombers, along with several smaller drones. [www.nasa.gov]
A popular enhancement on traditional wing design is the 'winglet.' When normally optimizing wing dimensions for a particular load and operating at some cruising speed, there are constraints set on wing length that relate to average gate sizes. Since extending the length of the wing allows for significantly more area and improves stability, this constraint is almost always active. To continue getting more wing "length" while allowing planes to fit in the airport gates, winglets were quickly introduced.
(http://en.wikipedia.org/wiki/Wing-shape_optimization) (http://web.mit.edu/drela/Public/papers/Pros_Cons_Airfoil_Optimization.pdf) (http://web.mit.edu/2.972/www/reports/airfoil/airfoil.html) (http://www.coe.psu.edu/water/images/b/b5/Alpman.pdf) (http://www.sciencedirect.com/science/article/pii/S1018363913000159) (https://www.grc.nasa.gov/www/k-12/problems/Lorri/Design_Airfoil_int.htm) (http://wright.nasa.gov/airplane/Images/liftold.gif) (http://www.grc.nasa.gov/WWW/k-12/airplane/wrights/Images/smeaton.gif) (http://adg.stanford.edu/aa241/airfoils/airfoilhistory.html)