Real-coded adaptive range genetic algorithm applied to transonic wing optimization
Introduction
Most of commercial aircrafts today, such as B747, B777, and A340 cruise at transonic speeds, that is, just below the speed of sound. During the long duration of cruise, engine thrust is applied to maintain aircraft speed against aerodynamic drag. Since a large part of their maximum takeoff weights is occupied by the fuel weight, the objective of an aerodynamic design optimization of a transonic wing is, in principle, minimization of drag.
Unfortunately, drag minimization has many tradeoffs. There is a tradeoff between drag and lift because one of the drag components called induced drag increases in proportion to the square of the lift. A wing that achieves no induced drag would have no lift. Another tradeoff lies between aerodynamic drag and wing structure weight. An increase in the wing thickness allows the same bending moment to be carried with reduced skin thickness with an accompanying reduction in weight. On the other hand, it will lead to an increase in another component of the drag called wave drag. Therefore, the aerodynamic design of a transonic wing is a challenging problem.
Furthermore, optimization of a transonic wing design is difficult due to the following factors. First, aerodynamic performance of a wing is very sensitive to its shape. Very precise definition of the shape is needed and thus its definition usually requires more than 100 design variables. Second, function evaluations are very expensive. An aerodynamic evaluation using a high fidelity model, such as the Navier–Stokes equations usually requires 60–90 min of CPU time on a vector computer.
Among optimization algorithms, gradient-based methods (GMs) are well-known algorithms, which probe the optimum by calculating local gradient information. Although GMs are generally superior to other optimization algorithms in efficiency, the optimum obtained from these methods may not be a global one, especially in the aerodynamic optimization problem.
On the other hand, genetic algorithms (GAs) are known to be robust methods modeled on the mechanism of natural evolution. GAs have capability for more global search because they don’t use any derivative information and they search from multiple design points. Therefore, GAs are a promising approach to aerodynamic optimizations.
Finding a global optimum in the continuous domain for the aerodynamic design is challenging even for GAs. In traditional GAs, binary representation has been used for chromosomes, which evenly discretizes a real design space. Since binary substrings representing each parameter with a desired precision are concatenated to form a chromosome for GAs, the resulting chromosome encoding a large number of design variables for real-world problems would result in a string length too long. In addition, there is discrepancy between the binary representation space and the actual problem space. For example, two points close to each other in the real space might be far away in the binary-represented space. It is still an open question to construct an efficient crossover operator that suits to such a modified problem space.
A simple solution to these problems is the use of floating-point representation of parameters as a chromosome [1]. In these real-coded GAs, a chromosome is coded as a finite-length string of the real numbers corresponding to the design variables. The floating-point representation is robust, accurate, and efficient because it is conceptually closest to the real design space, and moreover, the string length reduces to the number of design variables. It has been reported that the real-coded GAs outperformed binary-coded GAs in many design problems [2]. However, even the real-coded GAs would lead to premature convergence when applied to aerodynamic shape designs with a large number of design variables.
The objective of the present work is to develop robust and efficient GAs applicable to aerodynamic shape designs. To achieve this goal, the idea of dynamic coding, in particular adaptive range GAs [3], [4], is incorporated with the used of the floating-point representation. The resulting approach is then applied to a practical wing design problem as well as a simple test case to examine its performance.
To perform the practical wing design, the computation was processed in parallel using numerical wind tunnel (NWT) at National Aerospace Laboratory, Japan. NWT has 166 vector processing elements at peak performance of 280 GFLOPS. The actual computation took 108 h with 64 PE’s.
Section snippets
Adaptive range genetic algorithms
To treat a large search space with GAs more efficiently, sophisticated approaches have been proposed, referred to as dynamic coding, which dynamically alters the coarseness of the search space. In [5], Krishnakumar et al. presented Stochastic genetic algorithms (Stochastic GAs) to solve problems with a large number of real design parameters efficiently. Stochastic GAs have been successfully applied to flight propulsion controller designs [5] and air combat tactics optimization [6].
Adaptive
Aerodynamic design of a transonic wing
A wide range of approximations can represent the flow physics. Among them, the Navier–Stokes equations provide the state-of-the-art of aerodynamic performance evaluation for engineering purposes. Although the three-dimensional Navier–Stokes calculation requires large computer resources to estimate wing performances within a reasonable time, it is necessary because a flow around a wing involves significant viscous effects, such as potential boundary-layer separations and shock wave/boundary
Summary
To develop GAs applicable to practical aerodynamic shape designs, the real-coded ARGAs have been developed by incorporating the idea of the binary-coded ARGAs with the use of the floating-point representation. The real-coded ARGA has been applied to a practical aerodynamic design optimization of a transonic wing shape for generic transport as well as a simple test case. The test case result confirms the present GA outperforms the conventional GA.
Aerodynamic optimization was performed with 87
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- 1
Present affiliation: NASA Glenn Center, Cleveland, OH, USA.
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