# Airfoil Design Using Genetic Algorithms

Osama A. Marzouk

► To cite this version:

Osama A. Marzouk. Airfoil Design Using Genetic Algorithms. The 2007 International Conference on Scientific Computing (CSC’07), The 2007 World Congress in Computer Science, Computer Engineering, and Applied Computing (WORLDCOMP’07), World Academy of Science, Massachusetts Institute of Technology (MIT); Harvard University; University of Texas at Austin; Purdue University; Russian Academy of Sciences, Jun 2007, Las Vegas, Nevada, USA, United States. pp.127-132. hal-04862887

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# Airfoil Design Using Genetic Algorithms

Osama A. Marzouk

Department of Engineering Science and Mechanics, MC 0219
Virginia Polytechnic Institute and State University
Blacksburg, VA 24061 USA

**Abstract:** Genetic algorithms (GAs) are a search method used in finding true or approximate solutions to optimization and search problems. They have been applied in different fields like computer science, engineering, and economics. Here, we apply GA to the problem of designing a symmetric airfoil with high lift coefficient. Each population member, or chromosome, is an array representing the geometry of the airfoil. The optimization technique is applied to an initial population of airfoils having the geometry of NACA 4-digit series with different thickness ratios. Different GA operators are used including reproduction, binary encoding, crossover, and decoding. The flow field around the airfoil is solved using the finite difference method. The computational fluid dynamics (CFD) solver and the GA technique are combined together and yielded an optimal airfoil under certain optimization parameters. Different issues that arise from this process are discussed, like the necessity to add constraints to avoid unacceptable airfoil shapes.

**Key Words:** genetic algorithms, airfoil, aerodynamics

## 1 Introduction

The concept of evolutionary computing goes back to the 1960s. It was first introduced in the work of Rechenberg and then developed by others. Later, genetic algorithms were later introduced by Holland and developed by him and his students and colleagues.

Genetic algorithm was used by Koza [9] to evolve programs that perform certain tasks. He called his method “genetic programming.” LISP programs were used, because programs in this language can expressed in the form of a “parse tree”, which is the object the GA works on.

De Falco et al. [3] applied a modified version of GAs to the optimization of the performance of airfoil sections in a transonic flow, so that shock waves are eliminated. These shock waves incur shock drag and

unfavorable performance. As an adaptation to the standard GA, members of the last generation that have high fitness are forced to be in the new generation, while the rest are subjected to crossover and mutation. This approach has been also followed in our study as it enhances the convergence of the overall process. The fitness function was chosen to be proportional to the square of the lift coefficient and inversely proportional to the wave drag coefficient. These coefficients were obtained via solving the potential flow around the airfoil. The used method allowed finding shock-free airfoil designs with improved convergence rates.

Jones et al. [8] used GA to design rotor blades operating with high lift-to-drag ratios and at low noise. Some constraints were imposed on the minimum lift and pitching moment coefficients, so that the performance of any generated blade does not go below that of a NACA 0012 section. The lift and drag were obtained by an aerodynamic solver (XFOIL), which is based on integral boundary layer solution coupled with the linear vorticity-stream function panel method. The noise level was predicted by an aeroacoustic solver (WOPWOP). Both solvers were combined so that the flow field from the aerodynamic solver is post-processed by the aeroacoustic solver to predict the acoustic level.

Tse and Louis [15] considered the fitness evaluation phase of GAs since it consumes a considerable fraction of the computations, especially in aerodynamic problems where time-consuming computational fluid dynamics (CFD) methods are needed for evaluating performance. Therefore, they proposed two improved GA strategies to improve the performance of basic GA, which are micro genetic algorithms and artificial neural networks. They implemented the two methods to the inverse and direct airfoil design cases. They found that the proposed methods yielded some improvement in the overall design process.

Quagliarella and Vicini [14] used single and multiobjective GA for the optimization of a high-lift airfoil. Two flow solvers were coupled with the GA. The first
flow solver is based on the viscous-inviscid interaction, which solves the Euler equations in the inviscid region and the integral boundary layer formulation in the viscous region. The second flow solver is based on the full potential flow theory. The first model was used for high-lift configurations, while the second was used to optimize transonic performances for cruise configurations.

Hacioglu [5] coupled GA with artificial neural network to propose a new search method. He utilized this method in problems of inverse airfoil design. Instead of predicting CFD computations of a candidate airfoil, a well-designed neural network is used for predicting the candidate itself. At each step of the genetic process, the target pressure distribution is fed to the neural network. This produces a candidate solution (airfoil) of the inverse design problem. Adding this candidate airfoil to the population at each step could enhance the genetic process.

Pehlivanoglu and Hacioglu [13] considered the inverse design of airfoils in different flow conditions, where a geometry search was conducted to satisfy a desired pressure distribution. An improved vibrational genetic algorithm (VGA), developed by Hacioglu [4], was used to improve the convergence rate of the procedure. The VGA method is known to require smaller population sizes and shorter generation cycles. The potential flow was solved using vortex-panel method. The mesh and problem domain were chosen as structured O-type. Pseudo-time marching the governing equations using explicit 6-step Runge-Kutta integration was used to obtain the steady-state flow field. The fitness function was formulated as the reciprocal of the integrated difference between the aimed pressure distributions on the airfoil surface and the current ones.

Here, we apply GAs to find an optimum airfoil for a high lift coefficient in low speeds at small angles-of-attack. The flow is governed by the potential equations, and is solved using finite difference methods. The CFD solver is combined with the GA routine and certain parameters and constraints were used in the process.

## 2 The Flow Solver

The Laplace equation for the stream function, $\psi$, governs the flow field.

$$ \frac{\partial^2 \psi}{\partial x^2} + \frac{\partial^2 \psi}{\partial y^2} = 0 \qquad (1) $$

Where

$$ u \equiv -\frac{\psi}{\partial y} \qquad (2) $$

$$ v \equiv \frac{\psi}{\partial x} \qquad (3) $$

Where $u$ and $v$ are the velocity components in the horizontal (x) and vertical (y) directions, respectively.

The equation is transformed to a body-fitted curvilinear coordinate system $(\xi, \eta)$ where the airfoil surface is aligned with the line $\eta = 0$. The advantages of this are to ease the application of the boundary conditions, and to avoid numerical problems that may arise if we directly discretize Equation (1) if there are any grid points having $\Delta x$ or $\Delta y = 0$. The grid points of the physical grid is transformed to a uniform, unit square, orthogonal computational grid with uniform $\Delta\xi$ and $\Delta\eta$. Figure 1 shows a typical view of the grid in the (x, y) system; the coordinates are normalized w.r.t. the chordline length, c. The grid size was chosen based on grid-sensitivity analysis.

Equation (1) is transformed to the new coordinate system and some manipulation is performed to have the final form in a conservative form. Second order finite difference is used.

The transformed form of Equation (1) is discretized and then solved using successive over relaxation technique (SOR). Upon solving for $\psi$, the velocity field is obtained using Equation (2). Then, the pressure field is obtained using Bernoulli equation, which for our case is reduced to:

$$ P_{tot} = P + \frac{1}{2} \rho V^2 \qquad (4) $$

Where $P_{tot}$ is the total (or stagnation) pressure, and $P$ is the local static pressure, and $V$ is the local total velocity, which is equal to $\sqrt{u^2 + v^2}$.

Although Equation (1) is linear, equation (4) is nonlinear and so, nonlinear effects are considered. Integrating the pressure coefficient over the airfoil surface gives the lift coefficient ($C_l$). According to d'Alembert's Paradox, the drag coefficient for this problem is zero.

We would like to end this section with a comment about the solver feasibility; the current CFD solver was compared to the experimental data for NACA airfoils and was found to be a feasible tool to predict the lift coefficient for small values of the angle-of-attack, $\alpha < 10$ and low-speed subsonic regime. This is the range where we use this solver in the current optimization process.
## 3 The GA Procedure

Different GA operators are used in the search procedure; starting from an initial population of airfoils that are based on NACA 4-digit series symmetric airfoils ranging from 0.1 to 0.18; so that we don't start from identical populations. Certain stations along the chord line were fixed (with more stations near the nose since the gradients are larger there). The corresponding y-coordinates of the upper surface at these stations (or the half-thickness values) will be the variable to be optimized (the chromosome). A sample airfoils from the initial population is shown in Figure 2. The CFD solver is applied to these airfoils and the fitness function (the lift coefficient) is evaluated for each one. The fitness function is decreased if one chromosome violates any of the geometric constraints; so that it will be eliminated in the next generation after the reproduction operator is applied. In the reproduction phase, members from the current generation are selected proportional to their fitness and a new generation of same number of members is generated, with members of high fitness are reproduced and those of lower fitness are eliminated. This can be thought of a filter, to expedite the convergence of the GA, and also to eliminate any undesirable members of the generation. Then, the chromosomes are encoded; we use binary coding where each y-coordinate is coded to a number of bits. But before this, the real decimal numbers are converted to integers by scaling to the range where y-coordinates vary. Then, the crossover phase is performed (with certain probability). The chromosomes are then decoded to the decimal form back, and analyzed by the CFD solver in a new GA iteration level. The loop stops when difference in the fitness of the optimal chromosomes in two successive generations lies within a prescribed tolerance.

## 4 Results and Discussions

The described technique is used to find an optimal symmetric airfoil for the flow at $\alpha = 5^\circ$. Five bits are used in the encoding phase to represent each y-coordinate. A probability of 50% was used in the crossover phase. The tolerance for the fitness convergence is 0.00001. Members from the final generation (which has the optimal airfoil) are shown in Figure 3. The optimal airfoils in the last two generations, the final one and the one before it, are given in Figure 4. We realized that we need to enforce certain geometric constraints to eliminate chromosomes that have high fitness, but are not feasible or accepted geometrically, as they don't follow the conventional shape of

airfoils, and so they are hard to manufacture. Sample of such cases are presented in Figure 5. A constraint is imposed on the slope of the airfoil surface, so that an airfoil with very steep slope (except near the nose), or with decreasing thickness followed by an increasing one will have their fitness decreased before the reproduction. This ensures the elimination of them from the next generation. Convergence is usually encountered in the third or even the second generation; which indicates the favorable convergence speed of the performed process.

## 5 Conclusions

Genetic algorithms were combined with a CFD solver to design high-lift symmetric airfoil. The half-thickness values of the airfoil at certain stations along chordline for an airfoil represent the chromosome to be optimized. The main advantage of the presented method is its significant convergence speed; this is ascribed to different reasons. They include the simple flow model used, which is still reasonable within the limitation of low speed, small angle-of-attack flow; the reproduction phase; using good parameters in the GA operators; and the good choice of the initial population to be used in the search technique. Different constraints were made to avoid solutions which have strange shapes, although have good fitness. The described approach can be extended in different aspects, including the consideration of cambered airfoils, and the use of more expensive flow solver.

## References

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[2] Crosby, J. L. *Computer Simulation in Genetics*. John Wiley, London, 1973.

[3] De Falco, Del Balio, R., Della Cioppa, A., and Tarantino, E. A Parallel Genetic Algorithm for Transonic Airfoil Optimisation. *IEEE Transactions*, 429-434.

[4] Hacioglu A. Using Genetic Algorithm in Aerodynamic Design and Optimization. *Ph.D. Thesis*, Technical University of Istanbul.

[5] Hacioglu A. Augmented Genetic Algorithm with Neural Network and Implementation to the Inverse Airfoil Design. *10th AIAA/ISSMO Multidisciplinary Analysis and Optimization Conference*, New York, 30 August - 1 September, 2004.
[6] Hoffmann, K. A. and Chiang, S. T. *Computational Fluid Dynamics for Engineers*. Engineering Education System, Texas, 1989.

[7] Holland, J. Genetic Algorithms and the Optimal Allocations of Trials. *Society for Industrial and Applied Mathematics (SIAM), Journal of Computing*, 2:88-105, 1973.

[8] Jones, B. R., Crossley, W. A., and Lyrintzis, A. S. Aerodynamics and Aeroacoustic Optimization of Airfoils via a Parallel Genetic Algorithm. *AIAA 98-4811*, 1998.

[9] Koza, J. R. *Genetic Programming: On the Programming of Computers by Means of Natural Selection*. MIT Press: Cambridge, Massachusetts, 1992.

[10] Koza, J. R. Survey of Genetic Algorithms and Genetic Programming. *Proceedings of the WESCON 95*, 589-594, San Francisco, California, 7-9 November, 1995.

[11] Obayashi, S., Tsukahara, T., and Nakamura, T. Cascade Airfoil Design based Multiobjective Genetic Algorithms. *Genetic Algorithms in Engineering Systems: Innovations and Applications*, 446:24-29, San Francisco, California, 2-4 September, 1997.

[12] Obayashi, S., Tsukahara, T., and Nakamura, T. Multiobjective Genetic Algorithm Applied to Aerodynamic Design of Cascade Airfoils. *IEEE Transactions on Industrial Electronics*, 47(1):211-216, 2000.

[13] Pehlivanoglu, Y. V., and Hacioglu, A. Inverse Design of 2-D Airfoil via Vibrational Genetic Algorithm. *Journal of Aeronautics and Space Technologies*, 2(4):7-14, 2006.

[14] Quagliarella, D., and Vicini, A. Viscous Single and Multicomponent Airfoil Design with Genetic Algorithms. *Finite Elements in analysis and Design*, 37:365-380, 2001.

[15] Tse, D. C. and Chan, L. Y. Application of Micro Genetic Algorithms and Neural Networks for Airfoil Design Optimization. *Report by the National Research Council of Canada*, Institute for Aerospace Research, Ottawa (Ontario), 2000.
<figure><img src="image_2.png" /><figcaption>Figure 1: Illustration of the Grid Pattern.</figcaption></figure>

<figure><img src="image_3.png" /><figcaption>Figure 2: Members from the Initial Population.</figcaption></figure>
<figure><img src="image_4.png" /><figcaption>Figure 3: Members from the Final Generation.</figcaption></figure>

<figure><img src="image_5.png" /><figcaption>Figure 4: Optimum Airfoils in the Final Generation and the One Before.</figcaption></figure>

<figure><img src="image_6.png" /><figcaption>Figure 5: A Group of Airfoils with Unacceptable Members.</figcaption></figure>