Wing Weight Function

import numpy as np
import matplotlib.pyplot as plt
import uqtestfuns as uqtf

The Wing Weight test function [FSK08] is a 10-dimensional scalar-valued function. The function has been used as a test function in the context of metamodeling [ZZP+20] and optimization [FSK08].

Test function instance

To create a default instance of the wing weight test function:

my_testfun = uqtf.WingWeight()

Check if it has been correctly instantiated:

print(my_testfun)

Function ID      : WingWeight
Input Dimension  : 10 (fixed)
Output Dimension : 1
Parameterized    : False
Description      : Wing weight model from Forrester et al. (2008)
Applications     : metamodeling, sensitivity

Description

The weight of a light aircraft wing is computed using the following analytical expression:

\[ \mathcal{M}(\boldsymbol{x}) = 0.036 \, S_w^{0.758} \, W_{fw}^{0.0035} \, \left( \frac{A}{\cos^2{(\Lambda)}} \right)^{0.6} q^{0.006} \lambda^{0.04} \left(\frac{100 t_c}{\cos{(\Lambda)}}\right)^{-0.3} \left( N_z W_{dg} \right)^{0.49} + S_w W_p \]

where \(\boldsymbol{x} = \{ S_w, W_{fw}, A, \Lambda, q, \lambda, t_c, N_z, W_{dg}, W_p\}\) is the vector of input variables defined below.

Probabilistic input

Based on [FSK08], the probabilistic input model for the Wing Weight function consists of eight independent uniform random variables with ranges shown in the table below.

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print(my_testfun.prob_input)

Function ID     : WingWeight
Input ID        : Forrester2008
Input Dimension : 10
Description     : Probabilistic input model for the Wing Weight model from
                  Forrester et al. (2008).
Marginals       :

 No.    Name    Distribution    Parameters                 Description
-----  ------  --------------  -------------  -------------------------------------
  1      Sw       uniform       [150. 200.]             wing area [ft^2]
  2     Wfw       uniform       [220. 300.]      weight of fuel in the wing [lb]
  3      A        uniform        [ 6. 10.]              aspect ratio [-]
  4    Lambda     uniform       [-10.  10.]       quarter-chord sweep [degrees]
  5      q        uniform        [16. 45.]    dynamic pressure at cruise [lb/ft^2]
  6    lambda     uniform        [0.5 1. ]               taper ratio [-]
  7      tc       uniform       [0.08 0.18]   aerofoil thickness to chord ratio [-]
  8      Nz       uniform        [2.5 6. ]          ultimate load factor [-]
  9     Wdg       uniform       [1700 2500]      flight design gross weight [lb]
 10      Wp       uniform      [0.025 0.08 ]         paint weight [lb/ft^2]

Copulas         : Independence

Reference Results

This section provides several reference results of typical UQ analyses involving the test function.

Sample histogram

Shown below is the histogram of the output based on \(100'000\) random points:

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np.random.seed(42)
xx_test = my_testfun.prob_input.get_sample(100000)
yy_test = my_testfun(xx_test)

plt.hist(yy_test, bins="auto", color="#8da0cb");
plt.grid();
plt.ylabel("Counts [-]");
plt.xlabel("$\mathcal{M}(\mathbf{X})$");
plt.gcf().set_dpi(150);

Moments estimation

Shown below is the convergence of a direct Monte-Carlo estimation of the output mean and variance with increasing sample sizes.

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# --- Compute the mean and variance estimate
np.random.seed(42)
sample_sizes = np.array([1e1, 1e2, 1e3, 1e4, 1e5, 1e6, 1e7], dtype=int)
mean_estimates = np.empty(len(sample_sizes))
var_estimates = np.empty(len(sample_sizes))

for i, sample_size in enumerate(sample_sizes):
    xx_test = my_testfun.prob_input.get_sample(sample_size)
    yy_test = my_testfun(xx_test)
    mean_estimates[i] = np.mean(yy_test)
    var_estimates[i] = np.var(yy_test)

# --- Compute the error associated with the estimates
mean_estimates_errors = np.sqrt(var_estimates) / np.sqrt(np.array(sample_sizes))
var_estimates_errors = var_estimates * np.sqrt(2 / (np.array(sample_sizes) - 1))

# --- Do the plot
fig, ax_1 = plt.subplots(figsize=(6,4))

ax_1.errorbar(
    sample_sizes,
    mean_estimates,
    yerr=mean_estimates_errors,
    marker="o",
    color="#66c2a5",
    label="Mean",
)
ax_1.set_xlabel("Sample size")
ax_1.set_ylabel("Output mean estimate")
ax_1.set_xscale("log");
ax_2 = ax_1.twinx()
ax_2.errorbar(
    sample_sizes + 1,
    var_estimates,
    yerr=var_estimates_errors,
    marker="o",
    color="#fc8d62",
    label="Variance",
)
ax_2.set_ylabel("Output variance estimate")

# Add the two plots together to have a common legend
ln_1, labels_1 = ax_1.get_legend_handles_labels()
ln_2, labels_2 = ax_2.get_legend_handles_labels()
ax_2.legend(ln_1 + ln_2, labels_1 + labels_2, loc=0)

plt.grid()
fig.set_dpi(150)

The tabulated results for each sample size is shown below.

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from tabulate import tabulate

# --- Compile data row-wise
outputs = []
for (
    sample_size,
    mean_estimate,
    mean_estimate_error,
    var_estimate,
    var_estimate_error,
) in zip(
    sample_sizes,
    mean_estimates,
    mean_estimates_errors,
    var_estimates,
    var_estimates_errors,
):
    outputs += [
        [
            sample_size,
            mean_estimate,
            mean_estimate_error,
            var_estimate,
            var_estimate_error,
            "Monte-Carlo",
        ],
    ]

header_names = [
    "Sample size",
    "Mean",
    "Mean error",
    "Variance",
    "Variance error",
    "Remark",
]

tabulate(
    outputs,
    headers=header_names,
    floatfmt=(".1e", ".4e", ".4e", ".4e", ".4e", "s"),
    tablefmt="html",
    stralign="center",
    numalign="center",
)

Sample size	Mean	Mean error	Variance	Variance error	Remark
10	2.6046e+02	8.6249e+00	7.4389e+02	3.5067e+02	Monte-Carlo
100	2.6215e+02	4.8620e+00	2.3639e+03	3.3599e+02	Monte-Carlo
1000	2.6944e+02	1.5739e+00	2.4772e+03	1.1084e+02	Monte-Carlo
10000	2.6822e+02	4.8051e-01	2.3089e+03	3.2654e+01	Monte-Carlo
100000	2.6814e+02	1.5191e-01	2.3077e+03	1.0321e+01	Monte-Carlo
1000000	2.6815e+02	4.8135e-02	2.3170e+03	3.2768e+00	Monte-Carlo
10000000	2.6808e+02	1.5199e-02	2.3100e+03	1.0330e+00	Monte-Carlo

References

[FSK08] (1,2,3)

Alexander I. J. Forrester, András Sóbester, and Andy J. Keane. Engineering Design via Surrogate Modelling: A Practical Guide. Wiley, 1 edition, 2008. ISBN 978-0-470-06068-1 978-0-470-77080-1. doi:10.1002/9780470770801.

[ZZP+20]

Lavi R. Zuhal, Kemas Zakaria, Pramudita S. Palar, Koji Shimoyama, and Rhea P. Liem. Gradient-enhanced universal Kriging with polynomial chaos as trend function. In AIAA Scitech 2020 Forum. Orlando, Florida, 2020. American Institute of Aeronautics and Astronautics. doi:10.2514/6.2020-1865.