You may have heard that a particular new airplane was designed on the computer. Just what this means and what can or cannot be computed-aided is not obvious and while design and analysis methods are being computerized to a greater degree than was possible earlier, there are great practical difficulties in turning the design task entirely over to the computers.
The design process has, historically, ranged from sketches on napkins (Fig. 1) to trial, error, and natural selection (Fig. 2), to sophisticated computer-aided design programs (Fig. 3).
Figure 1. Aircraft concepts can start with very rough sketches, as did the human powered airplane, the Gossamer Condor.
Figure 2. Aircraft Design By Trial and Error
Figure 3. Computer-Aided Design of Aircraft
Because the process is so complex, involving hundreds or thousands of computer programs, many people at many locations, it is difficult to manage and companies are continuing to try to improve on the strategy. In the early days of airplane design, people did not do much computation. The design teams tended to be small, managed by a single Chief Designer who knew about all of the design details and could make all of the important decisions. Modern design projects are often so complex that the problem has to be decomposed and each part of the problem tackled by a different team. The way in which these teams should work together is still being debated by managers and researchers.
The goal of these processes, whatever form they take, is to design what is, in some sense, the best airplane. To do this requires that we address three basic issues:
1. What do we mean by best?
2. How can we estimate the characteristics of designs so we can compare two designs in a quantitative way?
3. How do we choose the design variables which yield an optimum?
The first of these questions is perhaps the most important one, for if we don't know what we are trying to achieve, or if we select the wrong goal, it doesn't matter how good the analysis method may be, nor how efficient is our optimization procedure. Nevertheless, this question is often not given sufficient attention in many optimization studies.
Defining the Objective
If we were to examine advertisements for aircraft it might seem that the definition of the best aircraft is very simple. Madison Ave. Aircraft Company sells the fastest, most efficient, quietest, most inexpensive airplane with the shortest field length. Unfortunately such an airplane cannot exist. As Professor Bryson puts it, "You can only make one thing best at a time." The most inexpensive airplane would surely not be the fastest; the most efficient would not be the most comfortable. Similarly, the best aerodynamic design is rather different from the best structural design, so that the best overall airplane is always a compromise in some sense (see Fig 4.). The compromise can be made in a rational way if the right measure of performance is used. Structural weight and lift to drag ratio, for example, become parts of a larger equation. The left hand side of this equation is termed the figure of merit or objective and depends on the intended application for the aircraft.
Figure 4. One can only make one thing best at a time.
Various quantities have been used for this purpose including those listed below. This list is applicable to commercial transport aircraft and is in order of increasing sophistication. Many studies of new aircraft currently use direct operating cost as a measure of performance. This quantity is a more representative measure of the aircraft's performance than is a number such as gross weight since it is sensitive to fuel costs and other important variables. While some estimate of fuel prices, depreciation rates, insurance, labor rates, etc. must be made in order to compute direct operating cost, it is not necessary to estimate airline traffic, fares, and other difficult-to-project variables which would be necessary for computing numbers such as profit or return on investment.
Possible measures of performance:
1. Minimum empty weight
2. Minimum take-off weight (includes some measure of efficiency as fuel weight is included)
3. Minimum direct operating cost (a commonly-used measure)
4. Minimum total operating cost (a bit more difficult to estimate)
5. Minimum system cost over X years (life-cycle cost)
6. Maximum profit
7. Maximum return on investment
8. Maximum payload per $ (Sometimes used for military aircraft)
Analyses and Modeling
Once we have decided on the definition of "best" we must find a way of relating the "design variables" to the goal. This process is shown schematically, below.
For aircraft design, this process is often extremely complex. The number of parameters needed to completely specify a 747 is astronomical. So one uses a combination of approximation, experience, and statistical information on similar aircraft to reduce the number of design variables to a manageable number. This may range from 1 or 2 for back-of-the envelope feasibility studies to hundreds or even thousands of variables in the case of computer-assisted optimization studies. Even when the situation is simplified the model is usually very complicated and difficult. One generally must use a hierarchy of analysis tools ranging from the most simple to some rather detailed methods.
Calculating the drag of even a simple wing is not just a matter of specifying span and area. Other parameters of importance include: taper, sweep, Reynolds number, Mach number, CL or alpha, twist, airfoil sections, load factor, distribution of bugs, etc.
This can be programmed and available as an analysis tool, but one must be very cautious. Which of these variables is included in the model? What if the wing is operating at 100,000 Reynolds number? Has it been compared with experiment in this regime?
As the design progresses, more information becomes available, and more refined analyses become part of the design studies. The expertise of a designer, these days, involves knowing what needs to be computed at what time and identifying the appropriate level of approximation in the analyses.
One of the most important, but least well understood parts of the design process is the conceptual design phase. This involves deciding on just what parameters will be used to describe the design. Will this be a flying wing? A twin-fuselage airplane? Often designers develop several competing concepts and try to develop each in some detail. The final concept is "down-selected" and studied in more detail.
Design Iteration and Optimization
The last question which must be addressed seems the most straightforward but is full of subtlety and potential pitfalls. There are several methods by which one chooses the design variables leading to the "best" design. All of these require that many analyses be carried out-often thousands of times. This requires that the model be simplified to the point that it is fast enough, but not to the point that it is worthless. (Einstein's saying comes to mind here: "Things should be as simple as possible, but no simpler.") When the design may be described by only a few parameters, the process is very simple. One investigates several cases, and usually can easily see where the optimum occurs. (Even this may be difficult if the computations are extremely time consuming and theories called 'design of experiments', 'response surfaces', and Taguchi methods are currently used to solve such problems.) When the number of variables is more than a few, more formal optimization is required. Two approaches to optimization are commonly used.
1) Analytic results: When the objective function can be represented analytically, it is sometimes possible to construct derivatives with respect to the design variables and produce a set of simultaneous equations to be solved for the optimum. The idea is that a necessary condition for an optimum (without constraints) is: dJ / dxi = 0 for all i. This approach is very useful for fundamental studies, but requires great simplification (often oversimplification). One can see how useful this is in example cases. Consider the determination of the CL for maximum lift to drag ratio, L/D. If we write: CD = CDp + CL^2 / AR
and L/D = CL / CD , then L/D is maximized when CD /CL is minimized
or (CD /CL)/CL= 0.
This implies that: 0 = (CDp/CL+ CL / AR) / CL = -CDp/CL2 + 1 / AR.
The result is that at maximum L/D: CDp = CL2 / AR. That is, the zero-lift drag is equal to the lift-dependent drag. This simple result is very useful, but one must be careful that the analysis is applicable. When the aspect ratio or CDp is very high, the drag departs from the simple model at the computed optimal CL. When the problem involves constraints, the derivative is not zero at the optimum, but a similar analytic approach is possible by introducing Lagrange multipliers, l. In such a case, when the constraints are represented by gi = 0 the condition for an optimum is: d(J + lj gj ) / dxi = 0 and gi = 0.
2) Numerical optimization: In most aircraft design problems, the analysis involves iteration, table look-ups, or complex computations that limit the application of such analytical results. In these cases, direct search methods are employed. The following are schemes that have been used in aircraft design:
a. Grid searching: A structured approach to surveying the design space in which designs are evaluated at points on a grid. The disadvantage with this approach is that as the number of variables increases, the number of computations increases very quickly. If one evaluated designs with just five values of each parameter, the number of computations would be 5n where n is the number of design variables. Note that when n = 10, we require almost 10 million design evaluations.
b. Random searches: A less structured approach that does not require as many computations as the design variables increase, is the random search. It also does not guarantee that the best solution will be found. This method is sometimes used after some of the more sophisticated methods, described below, have gotten stuck.
c. Nonlinear Simplex or Polytope Method: In this case, n+1 points are evaluated in an n-dimensional design space. One moves in the direction of the best point until no improvement is found. At that point, the distance between points is reduced and the method tries to refine the search direction. This method is described in more detail in the book, "Numerical Recipes". It is very simple and robust, but very inefficient when one must consider more than a few design variables. Nevertheless, it has been used in aircraft optimization.
d. Gradient methods: These methods involve computation of the gradient of the objective function with respect to the design variables. The gradient vector points in the direction of the steepest slope. Moving in this direction changes the objective function most rapidly. Several forms of gradient methods are used. The most simple of these is the method of steepest descents in which the design variables are changed to move in the direction of the gradient. This method is usually modified to make it more robust and efficient. Variants on this theme include the conjugate gradient method and quasi-Newton methods that estimate values of the second derivatives (Hessian matrix) to improve the estimate of the best search direction. Most of these methods use the gradient information to establish a search direction and then perform a one- dimensional search in this direction.
So that's it. We just put it on the computer and press Return and out pops a 777, right?
Not really. Despite its obvious utility, numerical optimization seems to have been talked about a lot more than it has been used. It certainly is talked about a great deal. Prof. Holt Ashley gave the AIAA Wright Brothers Lecture in 1982. It was entitled, "On Making Things the Best -- Aeronautical Uses of Optimization". For this lecture, he surveyed the relevant literature and found 4550 papers on optimal control, 2142 on aerodynamic optimization, 1381 on structural optimization. A total of 8073 papers, along with surveys, texts, etc.. But Ashley had a hard time finding a single case where this formal procedure was employed by industry. In his paper he cites the results of an informal survey he conducted on the uses of optimization.
Typical responses included:
· From an aeronautical engineer, experienced in civil and aeronautical structures, "One of the reasons that I stopped work in optimization was my dismay ... that there were so very few applications."
· From a Dean of Engineering who has known the field for over a quarter century: "I do not recollect any applications."
· From a foremost specialist on synthesis with aeroelastic constraints, "I am sorry, but I don't really have any..."
· From a recently-retired senior design engineer, describing events at his aerospace company, "For fifteen years I beat my head against a stone wall ... The end was: formal optimization techniques were never used in aircraft design (even to this day!). The company was forced to use them in its subsequent ICBM and space programs."
A great deal has changed in the past decade, however, and optimization techniques are (only now) starting to become a standard tool for engineering design. Why has it taken so long for these methods to become well-used, and why, still, are the methods not used everywhere?
There are a host of reasons:
1) First, the analysis, itself, of a complete aircraft configuration is rather complex, even without the optimization. Program size and complexity are such that only very well-documented and well-maintained computer programs can be used. These programs are often written by many people (some of whom have retired) over many years and it is very difficult for an individual to know what the program can and cannot do. Many grandiose plans for completely integrated aircraft design systems have fallen by the wayside because they quickly become unmanageable.
2) Any analysis makes certain approximations and leaves certain things out. Optimizers, however, may not understand that certain considerations have been omitted. Optimizers are notorious for breaking programs. They exploit any weakness in the analysis if that will lead to a "better" answer. Even when the result appears reasonable, several difficult-to-quantify factors are often omitted: the compatibility with future growth versions for instance, or the advantages associated with fleet commonality. Moreover, optimums are, by definition, flat, so that leaving something out of the objective can cause large discrepancies in the answer - the optimum is never optimal. Some examples are shown in figures 5 and 6. These are examples in which real-life testing, rather than reliance on simulation, is critical.
3) Ruts, creativity, and local minima: New technology changes the assumptions, constraints, experience. An optimizer is limited to consider those designs that are described by the selected parameter set. Thus, an optimizer and analysis that was written to design conventional structures may not know enough to suggest the use of composites. An optimizer did not invent the idea of folding tips for a 777, nor would it create winglets, canards, active controls, or laminar flow, unless the programmer anticipated this possibility, or at least permitted the possibility, in the selection of design variables. (Figure 7.)
4) Noisy objective functions: When the analysis involves table look-ups or requires iterative intermediate computations, the objective function can appear to vary in a non-smooth fashion. This causes difficulties for many optimizers, especially those that require derivative information.
5) The dangers of sub-optimization: It is tempting to fix many design variables and select a few at a time to optimize, then fix these and vary others. This is known as partial optimization or sub-optimization and, while it makes each study more understandable, it can lead to wrong answers. One must be very careful about the selection of design variables and avoid partial optimization.
Figure 5. "Optimal" Flight Path for Landing a Sailplane - An example of what happens when the analysis does not include sufficient constraints.
Figure 6. "Optimal" Redesign of Cessna Cardinal. Optimizer has exploited simplified lateral stability constraints.
Figure 7. A Variety of Designs Not Likely Invented by an Optimizer
6) Finally, optimization is sometimes not needed as there are few feasible designs may exist. In aircraft design, problems are often constraint-bound. That is, the constraints, themselves dictate the values of the design variables. When many constraints are active at the optimum, the value of the gradient is not zero, and a modification the gradient methods are needed. One approach to constrained optimization is simply to add a penalty to the objective function when the constraints are violated. Such penalty function methods sometime work, but lead to rather difficult design space topologies and can cause problems for the optimizer. Often the constraints are visualized (at least as they affect up to two design variables) in a plot called a summary chart. Examples are shown on the following pages.
A variety of new approaches are being explored to avoid these difficulties. Improved software development environments reduce some of the problems of communication, maintenance, etc.. Simply changing the computer language (even from Fortran IV to Fortran 90) helps in understanding and maintaining the program. Artificial intelligence (AI) is being used in several ways to improve the efficiency of aircraft design. The ideas are beginning to be described in conference and journal papers on the subject. Watch for articles in Aerospace America, AIAA Journal of Aircraft, and similar publications.
Figure 8. Example Summary Chart Showing Constraints - Sometimes little room exists in design space once the constraints are satisfied.