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E. M. Tirado and D. Quagliarella
– Then, the statistic of interest T (x ∗
1 , x ∗
2 , . . . , x ∗
n ) is calculated for these new
samples.
This statistic will show a probability distribution of its own. Thus, from this
distribution, the confidence intervals of the risk functions, like VaR or CVaR, are
obtained. In other words, the evaluation of confidence intervals would require
repeated samples of a given population, but only one sample is available. Thus, the
bootstrap method treats the real sample as a population and the repeated samples
needed for confidence interval estimation are obtained by re-sampling it with
replacement.
Finally, it must be mentioned that, although very attractive for its simplicity,
the bootstrap technique has also several disadvantages, thoroughly discussed in
scientific literature. Maybe the main drawback is that the bootstrap samples are
related to the original (real) sample in the same manner that the original sample
is related to the unknown population. Hence, if the original population sample is
not sufficiently representative of the whole population features, then the confidence
intervals computed by bootstrap might be completely misleading.
13.5 Application Example
This section is aimed to give a simple but significant example of robust aerodynamic
design optimization problem focused on an airfoil in incompressible conditions
subject to geometric and aerodynamic constraints. The goal is the improvement of
the airfoil performance by changing its shape. When a robust version of this problem
is faced, an optimal solution that is less vulnerable with respect to uncertainties
in operating conditions and geometric shape is obtained. The baseline airfoil is
the NACA 2412. The design conditions assumed are Mach = 0 and Reynolds =
0.5 × 10 6 .
The airfoil performance is measured by a quantity of interest Q defined by
drag coefficient c d plus some constraints that are here considered as penalties.
Consequently, the robust optimization problem requires the minimization of the αCVaR of Q, with α set to 0.9. The equality constraints are the lift coefficient (c l ),
which is fixed to 0.5, and the maximum thickness (t), which is fixed to the 12% of
the airfoil chord (c). The inequality constraints are the trailing edge angle (T EA),
which must be greater than or equal to 13 ◦ , the leading edge radius (LER), that
must be greater than or equal to 0.7% of the chord, and the boundary layer transition
point on the airfoil lower surface (XT R LOW ) that cannot be located at x/c greater
than 0.95. A constraint on the pitching moment was not considered. In addition, an
ERROR variable is set to 1 when the solver does not converge. In summary, the
problem constraints are reported below:
E. M. Tirado and D. Quagliarella
– Then, the statistic of interest T (x ∗
1 , x ∗
2 , . . . , x ∗
n ) is calculated for these new
samples.
This statistic will show a probability distribution of its own. Thus, from this
distribution, the confidence intervals of the risk functions, like VaR or CVaR, are
obtained. In other words, the evaluation of confidence intervals would require
repeated samples of a given population, but only one sample is available. Thus, the
bootstrap method treats the real sample as a population and the repeated samples
needed for confidence interval estimation are obtained by re-sampling it with
replacement.
Finally, it must be mentioned that, although very attractive for its simplicity,
the bootstrap technique has also several disadvantages, thoroughly discussed in
scientific literature. Maybe the main drawback is that the bootstrap samples are
related to the original (real) sample in the same manner that the original sample
is related to the unknown population. Hence, if the original population sample is
not sufficiently representative of the whole population features, then the confidence
intervals computed by bootstrap might be completely misleading.
13.5 Application Example
This section is aimed to give a simple but significant example of robust aerodynamic
design optimization problem focused on an airfoil in incompressible conditions
subject to geometric and aerodynamic constraints. The goal is the improvement of
the airfoil performance by changing its shape. When a robust version of this problem
is faced, an optimal solution that is less vulnerable with respect to uncertainties
in operating conditions and geometric shape is obtained. The baseline airfoil is
the NACA 2412. The design conditions assumed are Mach = 0 and Reynolds =
0.5 × 10 6 .
The airfoil performance is measured by a quantity of interest Q defined by
drag coefficient c d plus some constraints that are here considered as penalties.
Consequently, the robust optimization problem requires the minimization of the αCVaR of Q, with α set to 0.9. The equality constraints are the lift coefficient (c l ),
which is fixed to 0.5, and the maximum thickness (t), which is fixed to the 12% of
the airfoil chord (c). The inequality constraints are the trailing edge angle (T EA),
which must be greater than or equal to 13 ◦ , the leading edge radius (LER), that
must be greater than or equal to 0.7% of the chord, and the boundary layer transition
point on the airfoil lower surface (XT R LOW ) that cannot be located at x/c greater
than 0.95. A constraint on the pitching moment was not considered. In addition, an
ERROR variable is set to 1 when the solver does not converge. In summary, the
problem constraints are reported below: