Colaboration:

• BAe
• burgeon

The airfoil design was performed to provide an automated section design capability and to provide recommendations for optimization in three dimensional wing design.

The airfoil was modelled with a separate Bezier curve for each of the upper and lower surfaces ( Wright and Holden 1998). This gives a convenient surface geometry representation with a relatively small number of defining parameters. The leading and trailing edge airfoil poles were held fixed to prevent the airfoil from drifting in space. The poles vary in both x and y directions to allow the shape of the airfoil to be adjusted smoothly. To ensure that the leading edge tangency condition at the leading edge was met, the x values of the two poles adjacent to the leading edge on the upper and lower surfaces were held fixed. In total, 42 design variables were used in the problem. The bounds on the design variables were such that the poles could move up to half the distance between adjacent poles at the starting position. This ensured that they did not cross, which could cause inflexions, undesirable aerodynamically.

In this optimization problem, the objective function, formulated as a weighted sum of the values of aerofoil drag coefficient at Mach 0.73 and Mach 0.76

is minimised for a fixed lift coefficient C_l =0.7, which was achieved by modifying the aerofoil angle of attack and a Reynolds number, based on aerofoil chord, Re2d = 30.0 ´ 10⁶. The objective function was non-dimensionalised to a value close to 1.0 after the first test optimization.

The BVGK (Ashill et al., 1987) aerodynamic prediction code used to calculate the section lift, drag and pitching moment characteristics is a field integral method, incorporating an inviscid full potential method, an engineering boundary layer method and a wave drag prediction method. The pitching moment was taken about the ¼ chord position and the transition position was fixed at 2 % chord on the upper and lower surfaces. Typical run times for the discretization of an airfoil section into 65 points on each surface was ~ 50 seconds on a SUN Sparc station 20.

The test problem presented here represents an airfoil redesign from a BAe research section, in which new thickness requirements are specified. These were that the upper section spar depth at 5 % chord was 13 % thinner, the forward spar was 4 % thicker, the rear spar was 6 % thinner and the 80 % chord thickness was 26% thicker than the starting airfoil. The value of the pitching moment was restricted as a larger pitching moment would incur a trim drag and possibly a weight penalty. There was also constraint on form factor at the upper surface trailing edge in order to prevent the boundary layer being too close to separation. For the purpose of comparison, the optimization process was undertaken using three methods: Powell's Direct Search (PDS) developed by Powell (1969), Genetic Algorithm (GA) developed by Keane (1994) and MARS (Toropov et al. 1993 and 1996, van Keulen and Toropov 1998). For the PDS and GA, the suitably non-dimensionalised constraints were applied via the use of penalty functions (Fiacco and McCormick 1968 and Keane 1994).

Aerodynamic codes sometimes fail to converge unexpectedly when an input geometry is not a conventional aerodynamic shape. This could be considered as domain-dependent calculability of the functions in the optimization problem. Therefore, optimization methods need to be selected which can accommodate this feature and some action must be taken when the code fails. In this optimization problem a high value of the penalty function was returned if the aerodynamic analysis method failed for both the PDS and GA methods. The PDS technique does not use gradients per se, but performs individual experiments in the design space and infer gradients. It should be noted that the quality of gradients can be affected by the domain-dependent calculability of the functions. When the GA was used, a low fitness value was allocated to unrealistic designs thus producing a robust technique well suited to use in such incomplete design spaces.

The results presented for each optimization method in Figures 1 to 3 will be (a) the geometry, with exaggerated vertical scale and the drag polars at (b) M = 0.73 and (c) M = 0.76. Dashed curves correspond to the initial configuration of the aerofoil and the solid curves correspond to the obtained configurations.

The results from the PDS method are presented in Figures 1a to 1c. Figure 1a shows the geometry after 900 evaluations. There are, however, 3 constraint violations. Figures 1b and 1c show that the drag polar is slightly improved at Mach 0.73 and worse at Mach 0.76, with the additional benefit of the increased trailing edge thickness of the new airfoil.

The results from the GA, using a population size of 300 and 24 generations (7200 evaluations total) are presented in Figures 2a to 2c. Figure 2a shows the small change made between starting and final geometry during the optimization process. There is not much modification, except at the trailing edge, where the new thickness criterion is being met. This result has no constraint violations. In the drag polars, Figures 2b and 2c show that although the Mach 0.73 result is about 5 counts worse, the Mach 0.76 result is improved by about 15 counts at the design point. An optimization run was performed in the time frame of half a week.

The MARS method offers potential for application to problems where computational times for single computations are long and the design space is relatively complex i.e. there are many design variables and constraints. It has been used in the past extensively in structural applications (although, so far, not in British Aerospace), and this is its first application to aerodynamic design. The MARS method used 50 function evaluations (or experiments, slightly more than the number of design variables) per plan (or step) and it took 14 steps to convergence, i.e. 700 evaluations in total. The history of the optimization process shows that the optimum design was found in 10 iterations and the satisfaction of the termination criterion took the remaining 4 iterations. If the functions at a point in the plan was not calculated, i.e. BVGK failed to converge, then an additional point, randomly chosen from the current search subregion, was added to the plan until the requested number of evaluations (50 in this case) had run successfully. The results are shown in Figures 3a to 3c. Figure 3a shows the modifications made to the geometry are small. The leading edge is rounder. The increased trailing edge thickness is achieved. The drag polars shown in Figures 3b and 3b show an improved polar above C_l = 0.6 at Mach 0.73 and above C_l = 0.65 at Mach 0.76.

This example demonstrates the potential for numerical optimization in commercial aircraft design. It provides an effective vehicle with which to apply aerodynamic design techniques. Many popular optimization techniques, especially gradient-based search methods, can have problems with constraint handling in the cases of domain-dependent calculability. GA's can find good solutions in parts of the design space not explored by gradient search techniques. They can handle functions of domain-dependent calculability well, but can be computationally expensive. The MARS method appears to be a good choice when large numbers of evaluations would be prohibitively expensive, and this method is able to handle the constraints with a prescribed degree of accuracy.

Acknowledgements

The authors wish to acknowledge the contribution of Dr. Stephen Rolston of BAe Airbus for his Bezier geometry representation software and his advice on wing design example. Professor Andy Keane of Southampton University provided the OPTIONS Optimization software suite to BAe and advice regarding how best to use it. Carren Holden was entirely funded by British Aerospace Sowerby Research Centre for this work under the BAe CFD Corporate plan and the BAe Airbus Advanced Optimised Wing Demonstrator program.

References