The aerodynamic drag minimization for wing-fuselage configurations was carried out by using the real-coded Genetic Algorithm (GA). The potential flow CFD solver KTRAN was used to obtain the aerodynamic parameters such as lift and drag coefficients. The optimization processes were applied to a series of wing sections in the wing span direction which are represented by B-spline curves . The actual values of the coordinates of the control nodes of the B-spline curves were designated as the design variables. The ONERA M6 wing with an axi-symmetric fuselage and the RAE wing axi-symmetric body model were optimized at the free stream Mach numbers M∞=0.84 and M∞=0.80, respectively. The corresponding drag coefficients were reduced by 51% and 13% for the given lift coefficients CL = 0.2893 and CL = 0.1592, respectively.
Among optimization techniques, Genetic Algorithm (GA) is attractive for aerodynamic design as it is capable of finding a global optimum compared to a gradient-based technique. It is a search algorithm based on the principles of the natural selection and genetics. In traditional GA, the binary representation of the design variables discretizes the real design space. Although such GA has successfully been applied to a wide range of optimization problems [2, 3], it suffers from some disadvantages when applying to a real problem involving a large number of design variables. One of them is a huge string length. For example, a problem with 100 design variables with a precision of six digits results in string length of about 2000. GA would perform poorly for such design problems. Another drawback comes from the discrepancy between the binary space representation and the actual problem space. Two optimization points, even close to each other in the actual space, are far from each other in the representation space. In the present study, the floating point representation of the design variables was used, where an individual is characterized by a vector of real numbers. This representation is accurate and efficient because it is conceptually closer to the real design space. Moreover, the string length is reduced to the number of design variables. To obtain a realistic shape, the design variables, which represent the geometry, must also satisfy the geometry constraints. For example, in the aerodynamic wing shape optimization, the geometry parameters such as wing span, sweep, taper and twist, chord, thickness, leading edge radius and trailing edge angle on the wing sections, must be limited to reasonable values. The aerodynamic shape optimization consists in determining the values of the design variables, which usually represent the geometry of the aerodynamic components, such that the objective function is minimized subject to the satisfaction of the aerodynamic constraints. Nowadays, CFD has matured to a point where it can be widely used as a tool for aerodynamic design. In the
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present study, a potential-based CFD solver KTRAN [4], which is suitable for transonic flows, was used. It can handle both isolated wing and wing-fuselage configurations. From the comparison between the calculation using this solver and the experimental data in Figure 1 for RAE wing axi-symmetric body model [5], it is found that this solver gives fairly good results, indicating that it is reliable to be used for the present study. In a previous work [6], the Genetic Algorithm (GA), coupled with the CFD solver KTRAN, was successfully applied to the aerodynamic drag minimization for the ONERA M6 isolated wing. However, for practical use, the optimization for the wing-fuselage configuration is needed. In the present study, the aerodynamic optimization for this configuration was carried out. The ONERA M6 wing was still used, but an axi-symmetric fuselage was added. The RAE wing axi-symmetric body model was also optimized, leading to a lower drag coefficient under the specified lift coefficient.
Genetic Algorithm Operations GA works on a coding of the design variables subject to certain performance constraints. In this study, the wing is represented by a series of sections from root to tip. A 6th order B-spline curve is used to represent each section of the wing. The actual values of the (x, y) coordinates of the control nodes for the B-spline curves are designated as the design variables (Figure 2). There are 8 control points for each of the lower and upper sides of the section profile. The fuselage and the original wing planform were not altered. The optimization procedure only modifies the wing section shapes. Generally, the optimization starts from a given initial shape of each section, which is precisely defined by the coordinates of a large set points. The first step is therefore to find the control points based on the initial shape coordinates by using the least square function method. In this analysis, a given population represents a number of wing-body configurations, whereby each configuration itself is regarded as a single chromosome. Usually, the initial population should be created randomly in order to guarantee that the global optimal can be found. In this study, the initial population was generated by mutation with randomly selecting the mutation point on the wing sections. As suggested in reference [7], the use of Micro-Genetic Algorithm (µGA) can facilitate fitness
convergence and enhance the algorithm’s capability to avoid local optima. The implication behind µGA is that with a small population size, the sub-optimal solution can be rapidly achieved in a cycle of GA operation. Then a new cycle of GA operation starts with the new population members generated from the sub-optimal member in the previous cycle of GA operation. In this study, this µGA technique was used. The population size is set to 10. There are 10 generations in a cycle of GA operation. Fitness evaluation is the basis for GA search and selection procedures. GA aims to reward individuals (chromosomes) with high fitness values and to select them as parents to reproduce offsprings. The purpose of optimization in this study is to reduce the drag of a wing-body configuration for a given lift. Therefore, the ratio of the lift (CL) and drag (CD) coefficients is used as the fitness value (objective function value). The CFD solver KTRAN calculates the ratio of CL/CD and sends it to GA, which uses it as the fitness value. For a member in a population, at least one CFD call is needed. Therefore, an enormous number of CFD calls are needed for the entire optimization. The parents in GA are chosen based on the Roulette wheel method where the probability of a parent being chosen is proportional to its fitness value. Each pair of parents produces one offspring (chromosome) by crossover. Then, mutation is applied to the offspring. After a new population is produced, the fitness of each member is compared to that of the parent generation and the best and the second best members in the generation are assigned to the new generation without crossover or mutation (elitism) [8]. Using this technique guarantees that the best member in all the populations will not be filtered out by using the GA operators during the optimization procedures. A simple one-point crossover scheme is applied. The crossover point is selected randomly. Figure 3 shows a kid wing section, demonstrating how the crossover operates on a wing section. Some design variables (control nodes) on the kid wing section are from the dad wing section (squares) and some from the mom wing section (crosses). The probability of the crossover is set at 80%, as the use of smaller values was observed to deteriorate the GA performance [9]. Mutation is carried out by randomly selecting a gene (control node) and changing its value by an arbitrary amount within a prescribed range (1% chord), as illustrated in Figure 4. As this change is applied to the selected node, its neighboring nodes are also adjusted so that the change in slope and curvature of the section profile will not be too abrupt. As discussed by Mantel et
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al. [10], a high mutation rate of 80% is chosen for better GA performance with real number coding. As mentioned in the previous section, geometry constraints must be imposed to obtain a realistic wing-body configuration. In the present study, on each section of the wing, constraints concerning the maximum thickness (>8% chord) and the range of the trailing edge angle (>5o, <20o) were imposed to avoid the wing section from becoming extremely thin. A minimal gap between the two control nodes at the trailing edge was also forced to prevent crossing of upper and lower surfaces.
CFD Solver For CFD based performance assessment, Bombardier-developed solver KTRAN [4] for full potential flow was used. It is based on the modified form of the classical transonic small perturbation equation given by Boppe [11]. It can handle both isolated wing and wing-fuselage configurations. The overall crude grid spans the entire computational domain. The fine grids are imbedded in the global crude grid to model the aircraft components such as wing and fuselage. The purpose of the fine grids is to provide detailed computation in regions where the flow field gradients are large and other flow details are of importance for numerical resolution. They improve the resolution of shock waves and the calculation of forces and moments, while the global crude grid provides a link between the fine grid solutions and the crude grid solutions. The lift, drag and pitching moment coefficients are calculated by integrating the load coefficients. The wing induced drag is calculated by a Fourier analysis of the spanwise lift distribution. The wave drag is then simply obtained as the difference between the pressure-integrated drag and the induced drag. In the present study, this flow solver was used to calculate the objective function (CL/CD) and send it to GA, which then uses it as a fitness value.
Results and Discussions First, a drag minimization study was carried out for the ONERA M6 wing with an axi-symmetric fuselage configuration. The free stream Mach numbers is M∞= 0.84. The lift coefficient was held constant at CL = 0.2893. The wing was represented by 7 sections in the spanwise direction where GA operations are applied. As mentioned earlier, for simplicity, the original fuselage and wing planform were not altered. The optimization procedure only modifies the wing section shapes. The angle of attack
is allowed to vary during the course of the optimization process. The convergence history of the computation is shown in Figure 5. It was noted that after about 500 CFD calls, the fitness value reached its converged value. It should be mentioned that the maximum fitness corresponds to the best member in each generation and the averaged fitness is related to the entire members in the generation. The trend of the fitness in this figure strongly shows that the optimum was approaching from one generation to another, demonstrating the reliability of the Genetic Algorithm. Figure 6 displays the original and optimized section shapes of the wing at four span wise locations: η = 0.0, 0.44, 0.65 and 1.0. Figure 7 gives the corresponding pressure distributions on each section. For the transonic potential flow past a wing, the total drag consists of three parts, shock wave drag, induced drag and pressure drag. It is noted from these figures that the computed results produce two shock waves on the upper surfaces of both original and optimized wings. However, the strength of shock wave close to the leading edge for the optimized wing is greatly reduced from its original value, while the strength of the second shock wave is similar to the original one, causing the reduction of the wave drag. On the wing tip section, the optimized section shape leads to the smaller pressure difference between the wing upper and lower surfaces, causing the reduction of the induced drag. At the rearward part of the middle span sections, the optimization leads to a greater loading of the wing. This would then compensate for the lift lost at the forward part of the wing sections in order to keep the lift coefficient constant. This kind of pressure distribution (load decrease at forward part and increase at rearward part) also contributes to the reduction of the pressure drag. All of these result in the drag coefficient decrease from 0.02161 for the original wing-body configuration to 0.01055 for the optimized configuration, which represents 51% reduction under the fixed lift coefficient CL = 0.2893. It should be noted that the pressure distribution on each wing section is subject to both streamwise and spanwise (3D effects) flow field influences. The Mach number distributions for both original and the optimized configurations are displayed in Figures 8. From the shock wave patterns produced on the upper wing surfaces it is noted that the optimization leads to weakening of the shock close to the leading edge and a decrease in the extent of supersonic flow region. The required free stream angle of attack for the original wing-body configuration was α = 3.156 degrees to create the lift coefficient CL = 0.2893, while for the optimized configuration the free stream angle of attack was required
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at α = 1.539 degrees while maintaining the lift at the same constant value. The RAE wing axi-symmetric body model was also optimized. The free stream Mach numbers was M∞=0.80. The lift coefficient was held constant at CL=0.1592. There are eight sections from the wing root to the tip. The section shapes and the corresponding pressure distributions at four span wise locations, η = 0.0, 0.40, 0.75 and 1.0, are displayed in Figure 9 and 10, respectively. There is no shock wave on the wing surfaces. The total drag is mainly from the pressure drag and the induced drag. Both of them were reduced by the optimization which changed the wing section shapes. The total drag coefficient was decreased from 0.008656 to 0.007517, which is 13% reduction under the fixed lift coefficient CL = 0.1592. Compared with the optimization for M6 wing-body configuration, the reduction of the drag is smaller. This demonstrates that the drag is caused mainly by the shock waves for a transonic flow. The pressure distributions on the wing root section showed the complex flow around this area. The Mach number distributions for both original and the optimized configurations are displayed in Figure 11. The required free stream angle of attack for the original configuration was α=1.928 degrees to create the lift coefficient CL=0.1592, while for the optimized configuration the free stream angle was required to be of α=1.668 degrees to keep the same lift coefficient.
Conclusions The Genetic Algorithm has been successfully applied for the aerodynamic optimization of M6 wing-body configuration and the RAE wing axi-symmetric body model. The drag coefficients are reduced by 51% and 13% under the given lift coefficients for the two configurations, respectively. The optimized wing section at the forward part causes the reduction of the wave drag, while the section at the tip causes the reduction of the induced drag. The loss in lift owing to lower suction pressures at the leading edge is balanced through an increase in aft loading. The free stream angle of attack is adjusted to a smaller value for the optimized configuration.
Acknowledgement The authors are grateful for the financial support and encouragement of the Department of National Defense.
Design for Cruise Performance Efficiency”, Transonic Perspective Symposium, Progress in Aeronautics and Astronautics, Vol. 81, D. Nixon ed., AIAA, New York, 1982, pp. 81-147.
2. Chan, Y.Y., “Applications of Genetic Algorithms to Aerodynamic Designs”, Canadian Aeronautics and Space Journal, Vol. 44, No. 3, September 1998, pp. 182-187.
3. Dasgupta, D., and Michalewicz, Z., “Evolutionary Algorithm in Engineering Applications”, Springer, 1997.
4. F. Kafyeke, “An Analysis Method for Transonic Flow about Three Dimensional Configurations”, Technical Report, Canadair Ltd., Montreal, Canada, 1986.
5. Treadgold, D. A., Jones, A. F. and Wilson, K. H., “Pressure Distribution Measured in the RAE 8ft x 6ft Transonic Wind Tunnel on RAE Wing ‘A’ in Combination with an Axi-Symmetric Body at Mach Numbers of 0.4, 0.8 and 0.9”, AGARD-AR-138.
6. Zhang, F., Chen, S. and Khalid, M., “Genetic Algorithm Application to Transonic Wing Optimization”, 10th Annual Conference of CFD Society of Canada, June 9-11, 2002, Windsor, Canada.
7. K. Krishnakumar, “Micro-Genetic Algorithms for Stationary and Non-Stationary Function Optimization”, SPIE Vol. 1196, Intelligent Control and Adaptive Systems, 1989, pp. 289-296.
8. Davis, L., “Handbook of Genetic Algorithms”, Van Nostrand Reinhold, New York, 1990.
9. D. Tse and Y.Y. Chan, “Multi-Point Design of Airfoils by a Genetic Algorithm”, 8th Annual Conference of the CFD Society of Canada, Montreal, Canada, June, 2000.
10. Mantel, B., Periaux, J., Sefrioui, M., Stoufflet, B., Desideri, J. A., Lanteri, S. and Marco, N., “Evolutionary Computational Methods for Complex Design in Aerodynamics”, AIAA-98-0222, 36th AIAA Aerospace Sciences Meeting & Exhibit, Jan. 12-15, 1998, Reno, NV.
11. C. W. Boppe, “Transonic Flow Field Analysis of Wing Fuselage Configurations”, NASA CR 3243, May 1980.
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