GLOBAL OPTIMIZATION ISSUES FOR TRANSONIC AIRFOIL DESIGN

GLOBAL OPTIMIZATION ISSUES FOR TRANSONIC AIRFOIL DESIGN Howoong Namgoong1, William A Crossley2, and Anastasios S Lyrintzis3 School of Aeronautics and Astronautics Purdue University West Lafayette, Indiana 47907-1282 ABSTRACT Airfoil design is very important for aircraft, because the airfoil can determine, to a large extent, the aircraft’s performance With the development of CFD techniques, many optimization algorithms have been applied to airfoil shape design, but the use of local optimization tools (gradient-based methods) may risk missing the best designs The objective of this research is to explore issues for global optimization of airfoil shapes in the transonic flow region by comparing a Genetic Algorithm (GA) with a Gradient-based optimization Method (GM) To determine the local optimum characteristics of the airfoil shape design space, different airfoil shapes are used as base or starting airfoils The results showed that the GA generated nearly the same shape regardless of the different initial base airfoils, which suggests that these shapes are approaching the global optimum However, the GM produces a different solution for each of the base airfoil shapes, suggesting that these results are local optima Comparisons of the generated airfoil shapes, the airfoil shapes’ performance, and the computational effort expended as a result of the GA and GM methods are also presented NOMENCLATURE Cd C d1 C d2 Cl C l1 fi M tE x xk ω ξi Coefficient of drag Coefficient of drag at design Mach number1 Coefficient of drag at design Mach number2 Coefficient of lift Design lift coefficient Shape functions Free stream Mach number Time for one evaluation of object function Airfoil coordinate Control point for shape functions Weighting factor for multipoint design Design variables INTRODUCTION In the early stages of the design process, the search for optimal airfoil shapes encompasses a broad range of possibilities Many different optimization strategies and techniques have been applied for aerodynamic design problems If the aerodynamic design space is smooth enough (e.g continuous first derivatives), Gradient-based Methods (GM) usually have performance advantages over their global optimization counterparts As an example, in the adjoint approach the gradient information at a single design point can be obtained with the equivalent of two flow calculations2 However, the aerodynamic performance of an airfoil is very sensitive to the surface geometry, and it is difficult to guarantee the convexness of the objective functions used in airfoil optimization Moreover, in transonic airfoil design, the objective function itself may be discontinuous due to shock waves One of the well-known concerns of using gradient-based Graduate Research Assistant, School of Aeronautics and Astronautics, Student Member AIAA Associate Professor, School of Aeronautics and Astronautics, Senior Member AIAA Professor, School of Aeronautics and Astronautics, Associate Fellow AIAA optimization techniques is that they conclude their search at a point where some form of the Kuhn-Tucker conditions are satisfied; however, these conditions only describe a local optimum point For airfoil design, this means that an airfoil shape found by a gradient-based optimizer is likely the locally optimal solution nearest to the initial airfoil shape Few papers have addressed this problem of encountering locally optimum shapes in the airfoil shape design space f2 f1 f3 f5 f4 f6 f7 f8 0.9 0.8 0.7 0.6 0.5 0.4 Recently, the Genetic Algorithm (GA) has emerged as viable (although more costly) alternative for airfoil optimization (Refs 4, 5, 6), because of its global nature However, direct comparisons between the two approaches (a GA and a GM) have been sparse Thus, the main focus of this paper is to investigate whether or not the design space has numerous local minima through the comparison of drag improvement of GA and GM method If numerous local optimum points occur, global optimization techniques will be an appropriate way to design airfoils despite the penalty of computational time increases Further, parallel and / or distributed computing can mitigate some of the computational expense associated with the GA 0.3 0.2 0.1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 0.1 0.075 0.05 y 0.025 -0.025 DESIGN VARIABLES -0.05 To pose the airfoil shape optimization problem, the design variables that control the geometrical shape of airfoils are needed In the approach used for this paper, shape functions7 are added to a baseline airfoil shape The design variables are multipliers that determine the magnitude of the shape function as it is added to the baseline shape Figure depicts the shape functions and their individual effects on a baseline NACA 0012 airfoil -0.075 0.5 0.75 x Figure Shape functions (top), NACA 0012 with each shape function applied (bottom) e(k ) = ln(0.5) ln(1 − x k ) , The y-coordinate position of the upper and lower surface of the airfoil are then described using the following equation: y ( x ) = y ( x) Base Airfoil + ∑ ξ i f i ( x ) 0.25 e(k ) = ln(0.5) ln( x k ) , k = 1,2 k = 3,4,5,6,7,8 (6) The control points for all shape functions used here are as follows: (1) (ξi: Design Variables, fi: Shape Functions, i =1, 16) x k = 0.06, 0.13, 0.2, 0.4, 0.6, 0.8, 0.87, 0.94 Eight shape functions are applied to the upper surface, and these same eight functions are applied to the lower surface, for a total of 16 shape functions OBJECTIVE FUNCTION FORMULATION f k ( x ) = sin[ π(1 − x) ], k = 1,2 (2) f k ( x ) = sin [πx e ( k ) ], k = 3,4,5 (3) f k ( x) = sin[ πx e ( k ) ], k = 6,7,8 (4) e(k ) (5) For a single point optimization, the objective function is chosen as the drag coefficient at specified Mach number that has required value for lift coefficient In the approach to be used in this paper, the lift curve slope of an airfoil shape is calculated using two flow solutions, and then the angle of attack corresponding to the lift is predicted This angle of attack is used as an input for the flow solver Single Point Objective Function arrangement and resolution is a compromise between accuracy and efficiency Maximize: F ( xi ) = −100 * Cd1 1.5 (7) Subject to: C l = C l1 , x imin ≤ x i ≤ x imax (8) 0.5 -0.5 -1 -1.5 -1 0.5 1.5 Figure C-grid scheme using 180 × 30 points In the case of a two-point shape optimization, the problem formulation is shown below utilizing a weighting factor ( ω ) in the objective function Figure shows the drag decrease with the increase in number of surface grid points for the NACA0012 airfoil at a Mach number in the subsonic region (M = 0.4) Figure compares the Euler solution at the design point with the AGARD10 experimental data to show the accuracy of solver A stronger shock closer to the airfoil trailing edge is predicted, as expected from an Euler solver Maximize: (9) Subject to: C l = C l1 , Cl = Cl2 , x imin ≤ x i ≤ x imax -0.5 Y Multi Point Objective Function F ( xi ) = −100 * [ω * C d1 + (1 − ω) * C d ] X Airfoils designed for a single flow condition generally have poor performance in flow at other than the design condition Some of the off-design performance problems of single point optimization are presented in Refs.7 and It is recognized that the single-point problem may not be highly applicable for general airfoil shape design, but this simple formulation allows investigation of the design space (10) 0.04 In this formulation, the intent is to minimize the drag coefficient at two different Mach numbers The choice of the weighting factors affects the contribution of each term in the objective function The weighting factors of used by Drela in his discussion of airfoil optimization appear to provide a rational objective function A constraint is imposed to ensure the desired lift coefficient is obtained in both Mach number conditions 0.035 0.03 Cd 0.025 0.02 0.015 0.01 EVALUATION OF OBJECTIVE FUNCTION 0.005 Because the transonic regime is of great interest for airfoils and airfoil shape design, an Euler-code is used as the objective function evaluator to account for the effect of shocks in the transonic region The scheme that is used in this research is the Implicit Upwind Finite Volume Scheme suggested by Roe Because this is an inviscid Euler code, the drag predicted for the airfoil is essentially the wave drag 0 20 40 60 80 100 120 Grid points on surface Figure “Residual” subcritical Euler drag coefficient as a function of grid points on airfoil surface (NACA 0012 airfoil, M=0.4, α=5.00) Figure shows the C-type grid system used for solution of the Euler equations The grid size is 181×30 110 grid points are located on the surface of airfoil A comparison with other grids showed that this grid Individuals with better fitness values are considered “more optimal”, so this fitness value must reflect both the objective of the design problem and any constraints imposed upon the design -1.5 -1 The GA employs selection, crossover and mutation operators to perform its search The selection routine performs the survival of the fittest function that allows better individuals to survive and serve as parents for the next generation of designs Crossover combines portions of chromosomes from the surviving parent designs to form the next generation of designs; combining features of good designs on average, but not always, results in better designs This gives the GA its optimization-like capability The mutation operator is used quite infrequently, as in nature, but this operator can mutate a binary bit in a chromosome to its opposite value (e.g “0” to “1”), which may introduce beneficial design traits that did not exist in the current population If the mutated trait is poor, the design with this mutation will be unlikely to survive This process transforms an initial population of randomly selected designs into a population of individuals that have “adapted” to their environment by becoming “more optimal” Additional details of the genetic algorithm can be found in several texts, like Ref Cp -0.5 0.5 1.5 Experiment Euler Analysis RAE2822 [M=0.74,α=3.19] 0.25 0.5 0.75 X Figure Pressure coefficient distribution of Euler prediction vs published experiment (RAE 2822 airfoil, M=0.74, α=3.19°) During their execution, the present optimization routines (GA or GM) using the shape function multiplier variables can generate airfoil shapes that violate geometric constraints, such as crossing upper and lower surfaces In that case a large penalty is added to the objective function without evaluation of the airfoil drag However, using a GA for design optimization is computationally expensive To overcome the computational time problem, the GA is adapted to a coarse-grain parallel implementation In this research, a Master-Slave type parallelization is applied to convert a serial GA into a parallel program A MIMD-type IBM SP2 and a Linux-Cluster machine were used for calculation following the basic approach of Ref 13 The GA algorithm is inherently parallelizable, because for each airfoil out of total airfoil population (which is usually several hundreds) the objective function evaluation (i.e the Euler solver) can be done in parallel independently of other airfoils To illustrate this, Figure shows the total wall-time for the parallel GA with the different number of processors The communication time is about 38.8% of wall-time when using 30 CPUs The total computational time for the test function evaluations in the Linux cluster machine (with nodes connected via a 100base-T Ethernet) decreases nearly exponentially as the processor number increases The convergence tolerance of the Euler-code is not varied during the iteration, and the maximum residual is reduced about four orders of magnitude An upper limit of 2000 flow solver iterations is set for each airfoil evaluation Each function evaluation required a wall clock time of approximately minutes (= tE) on a Linux cluster machine that has AMD Athalon 1.2GHz CPU PARALLEL GENETIC ALGORITHM Since its first descriptions, the GA has been applied to many engineering optimization problems 11,12 Based on Darwin’s “survival of the fittest” concept, the GA performs optimization tasks by “evolving” a population of highly fit designs over many generations A GA has the ability to search highly multimodal, discontinuous design spaces The GA also locates designs at, or near, the global optimum without requiring a good initial design point The GA represents design variables as strings of binary numbers, which serve as chromosomes Initially, the GA randomly generates a population of individuals After decoding the chromosome of each individual into the corresponding design variables, each design is analyzed to determine a fitness value   Computational wall-time (Sec.) 4500 4000 3500 3000 2500 Ideal Speed-Up Parallel GA 2000 1500 1000 7-1-1 Genetic Algorithm Results The input values for GA are described below The population size and mutation rate were selected using empirically derived guidelines for GAs using tournament selection and uniform crossover Seven bits represent each of the 16 shape function multipliers, for a total chromosome length of 112 bits 500 10 Number of CPUs 20 30 40  Population size: 448 Resolution: bit Design Variables: 16 Total Chromosome: 112 Variables limits: − 0.01 ≤ x i ≤ 0.01     Mutation probability: 0.0022 Elitism Tournament selection Uniform crossover     Figure Speed-up of parallel GA, computational wall time vs number of CPUs GRADIENT-BASED OPTIMIZATION METHOD If the objective function and constraints provide a unimodal convex domain and are also differentiable, a gradient-based optimization method can find the global optimum solution However, it is very hard to prove convexness and differentiability for general engineering design problems14 The transonic airfoil design problem is also difficult to characterize as convex and unimodal The algorithm is based on an elitist reproduction strategy, where members of the population that are evaluated most fit are selected for reproduction Using the shape function approach to represent changes in the airfoil shape requires a base airfoil, so all of the GA runs are associated with one of the base airfoils However, the initial generation of the GA is generated randomly, so the GA’s search does not begin with the base airfoil The best airfoil shape encountered in selected generations during a parallel GA run is shown in Figures 6-8 along with the base airfoil sections In each case, the GA was allowed to run for 90 generations with no other stopping criteria For this effort, the method of feasible directions, as provided by the CONMIN subroutines, 15 is used as the gradient-based optimizer In this research, different initial airfoil shape designs are used to check the consistency of optimum points found by the gradientbased method This can give some indication of multiple local minima appearing in the design space RESULTS AND DISCUSSION 7-1 Single-Point Optimization Results The pressure distribution plots in Figures 6-8 show that the newly designed airfoils not have strong shock waves and maintain the specified design lift coefficient For the single-point optimization problem, the objective is to minimize the drag when the free stream velocity is M=0.74 while producing a lift coefficient Cl=0.733 Three base airfoils are chosen from the database of Ref 16; these airfoils are the NACA 0012, the RAE 2822, and the Whitcomb supercritical airfoil The NACA 0012 is a subsonic, symmetric airfoil, while the RAE 2822 and Whitcomb are cambered airfoils originally designed to reduce wave drag in transonic flight conditions A summary of this single-point optimization problem is presented as follows:  M = 0.74 Base airfoils a) NACA0012 b) RAE2822 c) Whitcomb Super Critical Airfoil C l = C l1 = 0.733 0.3 -1.5 -1.25 0.25 -1 0.2 -0.75 NACA0012 Generation Generation 10 Generation 20 Generation 30 Generation 90 Y 0.1 -0.5 Cp 0.15 -0.25 0.05 0.25 Generation Generation 10 Generation 20 Generation 30 Generation 90 0.5 -0.05 0.75 0.25 0.5 0.75 1 X 0.25 0.5 X Base Airfoil [NACA0012] 0.75 Figure Best airfoil shapes (left) and pressure coefficient distributions (right) in selected generations of the GA using the NACA 0012 base airfoil 0.3 -1.5 -1.25 0.25 -1 0.2 -0.75 Y 0.1 -0.5 Cp RAE2822 Generation Generation 10 Generation 20 Generation 30 Generation 90 0.15 -0.25 0.05 0.25 Generation Generation 10 Generation 20 Generation 30 Generation 90 0.5 -0.05 0.75 0.25 0.5 0.75 0.25 X 0.5 X Base Airfoil [RAE2822] 0.75 Figure Best airfoil shapes (left) and pressure coefficient distributions (right) in selected generations of the GA using the RAE 2822 base airfoil -1.5 0.3 -1.25 0.25 -1 0.2 -0.75 Y 0.1 -0.5 Cp Whitcomb Generation Generation 10 Generation 20 Generation 30 Generation 90 0.15 -0.25 0.05 0.25 Generation Generation 10 Generation 20 Generation 30 Generation 90 Base Airfoil [Whitcomb SuperCritical] 0.5 -0.05 0.75 0.25 0.5 0.75 X 0.25 0.5 X 0.75 Figure Best airfoil shapes (left) and pressure coefficient distributions (right) in selected generations of the GA using the Whitcomb supercritical base airfoil Figure presents the convergence history of the GA In Figure 9, although the starting points are different, the fitness values are converging to about the same value as the generation increases This suggests that each run is approaching the same drag performance for the optimal airfoil shape In Figure 11 the pressure coefficients of the best airfoils after 90 generations are compared The upper surface pressure contours exhibit a very similar shape, whereas the lower surface has some variety, but still keeps the same design lift coefficient None of these pressure distributions indicate a strong shock; hence, the low predicted values of Euler drag -0.5 -0.6 Fitness (-Cd*100) -0.7 -0.8 -1.5 -0.9 -1.25 -1 -1 -0.75 -1.1 -1.2 Cp -0.5 Base Airfoil (RAE2822) Base Airfoil (NACA0012) Base Airfoil (Whitcomb) 20 40 Generation 60 80 0.25 0.5 Figure Best fitness value convergence history for all three GA runs Y 0.5 X 0.75 CONMIN uses the method of feasible directions to perform its search through the design space To provide an initial design for the search, setting all shape function multipliers to zero gives the base airfoil Figure 12 shows the convergence history of the CONMIN program using the NACA0012 base airfoil as the starting point The number of iterations for CONMIN to meet its convergence criterion is eight 0.05 -0.05 -0.1 Base Airfoil (NACA0012), Generation 90 Base Airfoil (RAE2822), Generation 90 Base Airfoil (Whitcomb), Generation 90 0.75 0.25 7-1-2 Gradient-Based Optimization Results 0.1 0.5 Figure 11 Pressure coefficient distributions for best airfoils from 90th generations of GA runs 0.15 0.25 Base Airfoil NACA0012 (90 Generation) Base Airfoil RAE2822 (90 Generation) Base Airfoil Whitcomb (90 Generation) 0.75 The final best airfoil shapes are compared in Figure 10 The upper surfaces of the airfoils are very close to each other, whereas there are some differences in the lower surfaces Because the objective is to minimize the wave drag, the upper surface is more important than lower surface for a lifting airfoil If we add the pitching moment as a constraint then the lower surface would be also important The similarity of the upper surface shapes suggests that the GA is indeed approaching the same “optimal” airfoil shape, and this final shape seems to be independent of the base airfoil -0.25 X Figure 10 Best airfoil shapes from generation 90 of all three GA runs -1.6 0.3 0.25 -1.65 0.2 -1.7 -1.75 Y -Cd*100 0.15 NACA0012 Iteration Iteration Final Iteration 0.1 0.05 -1.8 -1.85 -0.05 Base Airfoil [NACA0012] Iteration Number Figure 12 Convergence history for CONMIN with NACA 0012 as the starting shape 0.1 0.2 0.3 0.4 0.5 X 0.6 0.7 0.8 0.9 -1.5 -1.25 Figure 13 shows the change of airfoil shape and the change of pressure coefficient during the iterations from the base NACA 0012 airfoil shape Only small changes to the shape are made during the search The airfoil is modified to reduce the shock of airfoils, but a substantial shock still remains upon convergence Comparing with the GA solution (Figure 10) the CONMIN results show some reduction of the shock strength on the upper surface, but the GA results show a much higher reduction of the shock strength -1 -0.75 Cp -0.5 -0.25 0.25 0.5 NACA0012 Iteration Iteration Final Iteration 0.75 0.25 0.5 0.75 X Figure 13 Airfoil shape designs (top) and pressure coefficient distributions (bottom) generated during CONMIN search using NACA 0012 base airfoil In the case of RAE2822 and Whitcomb Super Critical Airfoils, CONMIN converged to almost the same airfoils as the base airfoils (Figures 14-15) These results were expected because both the RAE2822 and the Whitcomb airfoils were designed for transonic flow 0.3 -1.4 GA Multipoint (M=0.68) GM Multipoint (M=0.68) -1.2 0.25 -1 0.2 -0.8 -0.6 Y 0.15 -0.4 Y RAE2822 Final Iteration 0.1 -0.2 0.05 0.2 0.4 0.6 -0.05 0.8 0.25 0.5 0.75 0.25 0.5 0.75 X X Figure 14 Initial RAE 2822 airfoil shape and final shape generated by CONMIN -1.4 0.3 -0.8 GA Multipoint (M=0.74) GM Multipoint (M=0.74) -1.2 -1 -0.6 0.25 Y -0.4 0.2 -0.2 Y 0.15 0.2 Whitcomb Final Iteration 0.1 0.4 0.6 0.05 0.8 0 0.25 0.5 0.75 X -0.05 0.25 0.5 0.75 Figure 16 Pressure coefficient distributions for twopoint objective function results at M=0.68 (top) and M=0.74 (bottom) X Figure 15 Initial Whitcomb airfoil shape and final shape generated by CONMIN Figure 16 shows the result of two-point optimization using the GA and CONMIN The pressure coefficient distributions for these two airfoils are also plotted in Figure 17 to investigate the effect of twopoint design The GA results also showed less shock wave in both design Mach number Using the gradient-based search method, a different converged solution results from each different initial airfoil This supports the notion that the transonic airfoil design space has many local optimum design points, and the GM simply finds the local optimum nearest to the initial airfoil shape 7-2 Multi Point Optimization Results A two-point design case was tried to investigate the effects of the objective function selection The weighting factor in equation (9) is ω = , and the design Mach numbers are M=0.68 and M=0.74 while keeping the design lift coefficient equal to 0.733 for both Mach numbers different converged solutions with the change of initial base airfoils This result shows that the design space of transonic airfoil has numerous local optimum points However, when we used GA as an optimization tool we were able to obtain a quite similar solution (at least in the upper surface, which is the important one in this case) even though we started from totally different airfoils Therefore, this study verified that the GA can be used as a robust global optimization technique, even though the transonic airfoil design space has several local optima In addition, our results showed that the increased CPU time needed for the GA can be addressed with a parallelization strategy, which made it possible to use an Euler solver for fitness evaluations with fast turnaround times 0.3 0.25 0.2 Y 0.15 Base airfoil (NACA0012) GM multipoint (CONMIN) GA multipoint (90 Generation) 0.1 0.05 -0.05 0.25 0.5 0.75 X ACKNOWLEDGMENT Figure 17 GA and CONMIN results for two-point objective function formulation The first author was support by a Purdue Research Foundation (PRF) grant The calculations were performed on a 104-node cluster acquired by a Defense University Research Instrumentation Program (DURIP) grant 7-3 Computational Efficiency Comparison Table compares the drag results and computational time from each method All times are multiplied by tE (= time needed for one function evaluation using the Euler method) In the case of the NACA0012 base airfoil, the GA result has much smaller drag coefficient than GM result However, the GA needs about 250 times more computational time than the GM if a serial code is employed When a parallel GA with 45CPUs is used the difference reduces to only about times more than the GM Even though the GM used in this research is not the best method to solve this airfoil optimization problem, it is reasonable to say that the GA’s penalty for higher computational cost can be alleviated using parallelization REFERENCES Reuther J J et al., ”Constrained Multipoint Aerodynamic Shape Optimization Using an Adjoint Formulation and Parallel Computers,” AIAA Journal of Aircraft, Vol.36, No1, 1999, pp.51-86 Jameson A., ”Re-engineering the Design Process Through Computation,” AIAA Journal of Aircraft, Vol.36, No1, 1999, pp.36-50 Obayashi, S., and Tsukahara, T., “Comparison of Optimization Algorithms for Aerodynamic Shape Design,” AIAA Journal, Vol.35, No.8., Aug 1997, pp 1413-1415 CONCLUSION Holst L.T., and Pulliam H T., “Aerodynamic Shape Optimization Using Real-Number-Encoded Genetic Algorithm,” AIAA Paper 2001-2473, 2001 We have developed a GA based airfoil optimization strategy based on shape functions We have investigated transonic airfoil design using the GA and the GM, and employing an Euler solver for the shock wave drag prediction In case of the GM, we arrived at Table Comparisons of drag values and computational costs for GA and GM runs Method GA 0.00586 40320 NACA 0012 GM 0.01642 160 160 * tE GA 0.00576 40320 ~ 896 * tE GM 0.01602 157 157 * tE GA 0.00614 0.01085 40320 79 ~ 896 * tE 79 * tE RAE2822 Whitcomb GM Drag (Cd) 10 Number of function evaluation Parallel Computational time with 45 processors Base airfoil ~ 896 * tE 11 Goldberg, D E., Genetic Algorithms in search optimization and machine learning, Addison-Wesley, MA, 1989 Quagliarella, D and Della Cioppa, A., ”Genetic Algorithms Applied to the Aerodynamic Design of Transonic Airfoils,” AIAA Paper 94-1896-CP, 1994 12 Hajela, P., “Genetic Search - An Approach to the Nonconvex Optimization Problem,” AIAA Journal, Vol 28, No.7, Jul 1990, pp 1205-1210 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Center, Hampton, VA, 1991 Roe, P L., “Approximate Riemann solvers, parameter vectors and difference schemes,” Journal of Computational Physics, 43,1981, pp 357-72 16 Selig, M., “UIUC Airfoil Data Site,” [http://amber.aae.uiuc.edu/~m-selig/ads.html], accessed Feb 2002 10 “Experimental Data base for Computer Program Assessment”, AGARD advisory report, No.138, 1979 11 ... d ] X Airfoils designed for a single flow condition generally have poor performance in flow at other than the design condition Some of the off -design performance problems of single point optimization. .. Because the transonic regime is of great interest for airfoils and airfoil shape design, an Euler-code is used as the objective function evaluator to account for the effect of shocks in the transonic. .. gradient-based optimization method can find the global optimum solution However, it is very hard to prove convexness and differentiability for general engineering design problems14 The transonic airfoil design