Aerodynamic profile design using genetic algorithms and CFD.

Vilag, Valeriu ; Popescu, Jeni ; Petcu, Romulus 等

1. INTRODUCTION

Classical methods for aerodynamic profile design rely on previous experimental results and simplified mathematical models describing the phenomena. Since the development of Computational Fluid Dynamics (CFD) codes the obtained geometries are verified once more before new experiments and corrections to the original geometry are often identified. The current development of computer hardware made this first validation process shorter and shorter arising the idea to use this great resource in the actual design process. The problem consisting in the fact that classical methods described in the iterature pretend to obtain the best geometry and the hand adjustments made using experiments and CFD simulations shows that improvements can be obtained, is solved using an optimization process for the whole design phase. This optimization is regarded in the present paper like the evolution of the profile geometry towards the best fulfilling of its aerodynamic purpose.

2. GENETIC ALGORITHMS OVERVIEW

Genetic algorithms are optimization algorithms following the natural evolution model (Kalyanmoy, 1999). From an original population, through certain mechanisms inspired by the nature, there are obtained inheritors fighting for survival. After relatively small number of populations the solution converges to the best generation which cannot be further improved.

In order to use genetic algorithms the designer must define the architecture of an individual, similar to DNA, representing the sets of parameters, chromosomes, defining its unique features (Koehler, 1997). The great number of combinations of possible values of these parameters shows the necessity to use an efficient way to decide which one is the best.

3. PROFILE PARAMETRIZATION

A relative simple way for obtaining theaerodynamic profile is defined. Two parabola portions are used: one for the skeleton and on for the thickness distribution of the profile. The parabola portions are further computed in order to obtain usefull data for geometry construction under Computer Aided Design software (CAD) (Yang, 2006).

[FIGURE 1 OMITTED]

Six parameters define the aerodynamic profile. [A.sub.1,2] for the peaks of the parabolas, [B.sub.1,2] for the starting point of each parabola portion and [C.sub.1,2] for the end point of the parabola portion. [A.sub.1,] [B.sub.1] and [C.sub.1] are used for the skleton and the others for the thickness distribution. Figure 1 shows the positioning of the parameters defining the parabola chosen such as to cross (0,0) and (1,0) point.

The "x" coordinate of the END point is given in equation (1), meaning that the END is always on a greater "x" coordinate with respect to the START point. The parabola is defined using equation (2), which is used for computing all the points on it.

[X.sub.END] = B + C (1-B) (1)

y = 4 A(x - [x.sup.2]) (2)

Possible values for the parameters are:

[A.sub.1,2], [C.sub.1,2] [member of] {0.1; 0.2; 0.3; 0.4; 0.5; 0.6; 0.7; 0.8; 0.9; 1}

[B.sub.1,2] [member of] C {0; 0.1; 0.2; 0.3; 0.4; 0.5; 0.6; 0.7; 0.8; 0.9}

Each parabola portion is split into 25 parts of equal length between the START and the END points, the parabola portion defining the skeleton is scaled to a chord equal to 1m and then rotated to zero incidence and the parabola portion defining the thikness distribution is scaled such as to obtain a maximum given thickness and 24 pairs of points are obtained to be arranged symmetrical on the already obtained skeleton. In this stage, a CAD software is used to import the points for the suction and pressure side through which a spline curve is constructed from the trailing edge to the leading edge and back, as shown in Figure 2.

[FIGURE 2 OMITTED]

4. EVOLUTION

From an original population, which is arbitrarily chosen, there are two main techniques to obtain the inheritors fighting for survival (Temesgen, 2007). We chose the following two individuals as primordial profiles and we will show the crossover and the mutation.

#1 = {0.5; 0.5; 0.5; 0.5; 0.5; 0.5}

#2 = {0.6; 0.6; 0.6; 0.6; 0.6; 0.6}

The crossover can be done using one or multiple crossover point but for this example we chose crossover at the middle of the DNA sequence. We obtain the following inheritors:

#3 = {0.5; 0.5; 0.5; 0.6; 0.6; 0.6}

#4 = {0.6; 0.6; 0.6; 0.5; 0.5; 0.5}

If we compute the aerodynamic performances for the four individuals obtained so far we see the numbers presented in Table 1, where CL and CD are the computed lift and drag coefficients.

We can see that #1 is the best followed by #3. #2 and #4 didn't survive to give birth to new inheritors.

In order to increase the diversity of the population we can use the mutation evolution. This technique is also arbitrarily, and we chose to apply it for profile #1 by decreasing its second parameter. We obtain profile #5:

#5 = {0.5; 0.4; 0.5; 0.6; 0.6; 0.6}

The results for the new individual shows a [C.sub.L] / [C.sub.D] = 30.9784, meaning that it will take the place of #3 and will form the new generation along with #1.

5. CFD CONTOUR PLOTS

For profile #5, which is the best one so far, we show the relative pressure distribution and velocity distribution.

[FIGURE 3 OMITTED]

[FIGURE 4 OMITTED]

[FIGURE 5 OMITTED]

Figures 3 and 4 are ony qualitative ones showing that the profile can fulfill a certain aerodynamic purpose(Gelsey, 1998). The CFD code predicts the pressure and velocity distribution on the profile but in this paper we do not discuss its precission. We only add that the convergence is obtain in less then 1000 iterations taking about 4 minutes to compute.

6. TECHNIQUE UTILIZATION

The technique can be exploited for aerodynamic design problems where a 2D approach is sufficient by applying multiple times one of the evolution methods described in an iterative process. A large spectrum of profiles can be obtained and the optimization process will lead to the best profile for the imposed working conditions. Several profile types are displayed in Figure 5.

7. CONCLUSION

The presented technique, which combines CFD with genetic algorithms, can be used for optimization processes when designing 2D aerodynamic profiles. The parametrization allows to obtain many types of profiles meaning that many aerodynamical problems can be solved using this method. Several steps shall be undertaken in order to use the method for 3D designing problems where the number of parameters will increase and wings or blades can be generated and optimized.

8. REFERENCES

Gelsey, A.; Schwabacher, M. & Smith, D. (1998). Using modelling knowledge to guide design space search, In: Artificial Intelligence 101, pp 35-62, Elsevier

Kalyanmoy, D. (1999). An introduction to genetic algorithms Vol. 24, Sadhana, India

Koehler, G. (1997). New directions in genetic algorithm theory. Annals of Operations Research 75, pp 49-68, Springer

Temesgen, M. & Wahid, G. (2007). Aerodynamic optimization of turbomachinery blades using evolutionary methods and ANN-based surrogate models, Springer Science+Business Media, DOI 10.1007/s11081-007-9031-1

Yang, Y.; Wu, Y. & Liu, S. (2006). Graphical model construction based on evolutionary algorithms, Journal of Control Theory and Applications 4, pp 349-354, Springer

Tab. 1. Computed results for the first four individuals

Profile [C.sub.L] / [C.sub.D]

#1 30.4685
#2 20.4332
#3 30.2376
#4 19.8012