Learning Airfoil Parameters

This is a tutorial to determine the aerodynamic coefficients of a given airfoil using GENN in SMT (other models could be used as well). The obtained surrogate model can be used to give predictions for certain Mach numbers, angles of attack and the aerodynamic coefficients. These calculations can be really useful in case of an airfoil shape optimization. The input parameters uses the airfoil Camber and Thickness mode shapes.

• Inputs: Airfoil Camber and Thickness mode shapes, Mach, alpha

• Outputs (options): cd, cl, cm

In this test case, we will be predicting only the Cd coefficient. However, the other databases for the prediction of the other terms are available in the same repository. Bouhlels mSANN uses the information contained in the paper [1] to determine the airfoil’s mode shapes. Moreover, in mSANN a deep neural network is used to predict the Cd parameter of a given parametrized airfoil. Therefore, in this tutorial, we reproduce the paper [2] using the Gradient-Enhanced Neural Networks (GENN) from SMT.

Briefly explaining how mSANN generates the mode shapes of a given airfoil:

1. Using inverse distance weighting (IDW) to interpolate the surface function of each airfoil.

2. Then applying singular value decomposition (SVD) to reduce the number of variables that define the airfoil geometry. It includes a total of 14 airfoil modes (seven for camber and seven for thickness).

3. Totally 16 input variables, two flow conditions of Mach number (0.3 to 0.6) and the angle of attack (2 degrees to 6 degrees) plus 14 shape coefficients.

4. The output airfoil aerodynamic force coefficients and their respective gradients are computed using ADflow, which solves the RANS equations with a Spalart-Allmaras turbulence model.

See also [3], [4], [5], [6] for more information.

References

[1]

Bouhlel, M. A., He, S., & Martins, J. R. (2020). Scalable gradient–enhanced artificial neural networks for airfoil shape design in the subsonic and transonic regimes. Structural and Multidisciplinary Optimization, 61(4), 1363-1376.

[2]

Bouhlel, M. A., He, S., & Martins, J. R. (2019). mSANN Model Benchmarks, Mendeley Data, https://doi.org/10.17632/ngpd634smf.1.

[3]

Li, J., Bouhlel, M. A., & Martins, J. R. (2019). Data-based approach for fast airfoil analysis and optimization. AIAA Journal, 57(2), 581-596.

[4]

Bouhlel, M. A., & Martins, J. R. (2019). Gradient-enhanced kriging for high-dimensional problems. Engineering with Computers, 35(1), 157-173.

[5]

Du, X., He, P., & Martins, J. R. (2021). Rapid airfoil design optimization via neural networks-based parameterization and surrogate modeling. Aerospace Science and Technology, 113, 106701.

[6]

University of Michigan, Webfoil, 2021. URL http://webfoil.engin.umich.edu/, online accessed on 16 of June 2021.

Implementation

Utilities

import os
import numpy as np
import csv

WORKDIR = os.path.dirname(os.path.abspath(__file__))


def load_NACA4412_modeshapes():
    return np.loadtxt(open(os.path.join(WORKDIR, "modes_NACA4412_ct.txt")))


def load_cd_training_data():
    with open(os.path.join(WORKDIR, "cd_x_y.csv")) as file:
        reader = csv.reader(file, delimiter=";")
        values = np.array(list(reader), dtype=np.float32)
        dim_values = values.shape
        x = values[:, : dim_values[1] - 1]
        y = values[:, -1]
    with open(os.path.join(WORKDIR, "cd_dy.csv")) as file:
        reader = csv.reader(file, delimiter=";")
        dy = np.array(list(reader), dtype=np.float32)
    return x, y, dy


def plot_predictions(airfoil_modeshapes, Ma, cd_model):
    import matplotlib

    matplotlib.use("Agg")
    import matplotlib.pyplot as plt

    # alpha is linearily distributed over the range of -1 to 7 degrees
    # while Ma is kept constant
    inputs = np.zeros(shape=(1, 15))
    inputs[0, :14] = airfoil_modeshapes
    inputs[0, -1] = Ma
    inputs = np.tile(inputs, (50, 1))

    alpha = np.atleast_2d([-1 + 0.16 * i for i in range(50)]).T

    inputs = np.concatenate((inputs, alpha), axis=1)

    # Predict Cd
    cd_pred = cd_model.predict_values(inputs)

    # Load ADflow Cd reference
    with open(os.path.join(WORKDIR, "NACA4412-ADflow-alpha-cd.csv")) as file:
        reader = csv.reader(file, delimiter=" ")
        cd_adflow = np.array(list(reader)[1:], dtype=np.float32)

    plt.plot(alpha, cd_pred)
    plt.plot(cd_adflow[:, 0], cd_adflow[:, 1])
    plt.grid(True)
    plt.legend(["Surrogate", "ADflow"])
    plt.title("Drag coefficient")
    plt.xlabel("Alpha")
    plt.ylabel("Cd")
    plt.show()

Main