14 Best Practices for Surrogate Based Uncertainty Quantification in. . .
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acceleration of the flow and the shockwave at the single design condition, the
deterministic configuration decreases the curvature of the upper surface of the
airfoil. The robust configuration decreases it even further, in order to mitigate the
stronger shockwaves happening at higher lift coefficients and Mach numbers due to
the uncertainty. As a trade-off, the drag reduction will not be as good at lower Mach
and lift coefficients. As a result, the robust configuration is effective in reducing the
upper tail of the CDF of Fig. 14.11.
Table 14.3 summarizes the results of the deterministic and robust optimization.
The use of a robust approach further reduces the 95% quantile by 3.5 drag counts,
compared to the classical deterministic optimization.
14.5 Conclusions
Traditionally, the use of simulation-based design optimization in aerodynamics has
been carried out neglecting uncertainty, in a deterministic fashion. In practice,
deterministic optimization can lead to a mismatch between computational and
experimental results and serious performance losses due to aleatory uncertainties.
The uncertainties in the design conditions must be considered through a robust
optimization framework in order to come up with a realistic configuration that is
less sensitive to such uncertainties.
To deal with optimization under uncertainty, a statistical approach must be
followed in which the operating conditions and geometrical uncertainties are
modelled as samples of independent random variables. The objective function
becomes either a combination of statistical moments such as mean or standard
deviation of a quantity of interest or a quantile. In both cases, a full representation
of the stochastic space is necessary to evaluate the uncertainty.
A possibility to reduce the computational cost for an efficient uncertainty
quantification is the approximation of the output response through surrogate
models. Kriging is especially appropriate for this task as it allows the use of
scattered sampling, provides a measure of its prediction error, is convenient for
high dimensional problems, and allows the incorporation of gradients through
Gradient-Enhanced Kriging. This is specially attractive when the sensitivities can
be efficiently evaluated through a discrete adjoint method.
To validate the surrogate based uncertainty quantification for aerodynamic shape
optimization, the RAE2822 airfoil is optimized under operational uncertainties.
The Subplex optimizer is used and the statistics are evaluated at each iteration
using Kriging meta-models. The optimum configuration is able to reduce the 95%
quantile of the drag by 28% with respect to the baseline configuration. The optimum
configuration is effective in reducing the average strength of the normal shockwave
while having a larger longitudinal displacement along the airfoil. The framework is
able to provide an accurate estimation of the statistics at a reduced computational
cost.
451
acceleration of the flow and the shockwave at the single design condition, the
deterministic configuration decreases the curvature of the upper surface of the
airfoil. The robust configuration decreases it even further, in order to mitigate the
stronger shockwaves happening at higher lift coefficients and Mach numbers due to
the uncertainty. As a trade-off, the drag reduction will not be as good at lower Mach
and lift coefficients. As a result, the robust configuration is effective in reducing the
upper tail of the CDF of Fig. 14.11.
Table 14.3 summarizes the results of the deterministic and robust optimization.
The use of a robust approach further reduces the 95% quantile by 3.5 drag counts,
compared to the classical deterministic optimization.
14.5 Conclusions
Traditionally, the use of simulation-based design optimization in aerodynamics has
been carried out neglecting uncertainty, in a deterministic fashion. In practice,
deterministic optimization can lead to a mismatch between computational and
experimental results and serious performance losses due to aleatory uncertainties.
The uncertainties in the design conditions must be considered through a robust
optimization framework in order to come up with a realistic configuration that is
less sensitive to such uncertainties.
To deal with optimization under uncertainty, a statistical approach must be
followed in which the operating conditions and geometrical uncertainties are
modelled as samples of independent random variables. The objective function
becomes either a combination of statistical moments such as mean or standard
deviation of a quantity of interest or a quantile. In both cases, a full representation
of the stochastic space is necessary to evaluate the uncertainty.
A possibility to reduce the computational cost for an efficient uncertainty
quantification is the approximation of the output response through surrogate
models. Kriging is especially appropriate for this task as it allows the use of
scattered sampling, provides a measure of its prediction error, is convenient for
high dimensional problems, and allows the incorporation of gradients through
Gradient-Enhanced Kriging. This is specially attractive when the sensitivities can
be efficiently evaluated through a discrete adjoint method.
To validate the surrogate based uncertainty quantification for aerodynamic shape
optimization, the RAE2822 airfoil is optimized under operational uncertainties.
The Subplex optimizer is used and the statistics are evaluated at each iteration
using Kriging meta-models. The optimum configuration is able to reduce the 95%
quantile of the drag by 28% with respect to the baseline configuration. The optimum
configuration is effective in reducing the average strength of the normal shockwave
while having a larger longitudinal displacement along the airfoil. The framework is
able to provide an accurate estimation of the statistics at a reduced computational
cost.