424
E. M. Tirado and D. Quagliarella
Table 13.1 Optimization parameters for the optimization
Maximum evaluations
Population size
Initial standard deviation
7000
20
0.1
integral module. Moreover, the laminar to turbulent flow transition is predicted using
the e N method [17].
The optimization algorithm selected for solving the described design problem
is the Covariance Matrix Adaptation Evolution Strategy (CMA-ES) [8], which is a
stochastic optimization algorithm based on self-adaptation of the covariance matrix
of a multi-variate normal distribution. It is mainly used for design optimization
problems up to a few hundreds of design variables. The parameters used for
the optimization algorithm are the maximum number of allowed evaluations, the
population size λ, and the initial standard deviation σ . The parameters set for this
problem are reported in Table 13.1.
Furthermore, it was mentioned that in case of non-convergence of the solver an
ERROR flag was set equal to 1. When robust optimization is faced, the treatment
of these cases is crucial for the optimization process. The quantity of interest to
be minimized, CV aR, depends on the upper tail of the Cumulative Distribution
Function. As a consequence, assigning a high value to the objective in the cases
where convergence is not achieved implies a too high CV aR value that could be
detrimental for the optimization algorithm behaviour. Hence, a proper value of the
objective in these cases must be decided. In particular, in this optimization problem,
the worst objective value selected between the properly converged cases, is assigned
to those in which the error flag is set. Numerical tests lead to conclude that this was
the setup with lowest impact on the optimization process behaviour.
13.5.1 Results
The obtained results are here commented and compared with the baseline airfoil
and the equivalent deterministic solution of the optimization problem. Firstly, in
Fig. 13.3, the airfoil shape of the robust optimized airfoil (dotted line) is compared
with the baseline airfoil (solid line) and the deterministic optimized airfoil (dashed
line).
In addition, the Cumulative Distribution Function obtained by introducing
uncertainties in the airfoil shape are reported in Fig. 13.4 for the initial NACA 2412
airfoil (solid line), for the deterministic optimized airfoil (dashed line), and for the
robust optimized airfoil (dotted line).
The comparison of the CDFs related to the deterministic and robust optimized
airfoils highlights that the robust optimal solution is less vulnerable to uncertainties
in geometric shape with respect to the deterministic one. This can be also deduced
E. M. Tirado and D. Quagliarella
Table 13.1 Optimization parameters for the optimization
Maximum evaluations
Population size
Initial standard deviation
7000
20
0.1
integral module. Moreover, the laminar to turbulent flow transition is predicted using
the e N method [17].
The optimization algorithm selected for solving the described design problem
is the Covariance Matrix Adaptation Evolution Strategy (CMA-ES) [8], which is a
stochastic optimization algorithm based on self-adaptation of the covariance matrix
of a multi-variate normal distribution. It is mainly used for design optimization
problems up to a few hundreds of design variables. The parameters used for
the optimization algorithm are the maximum number of allowed evaluations, the
population size λ, and the initial standard deviation σ . The parameters set for this
problem are reported in Table 13.1.
Furthermore, it was mentioned that in case of non-convergence of the solver an
ERROR flag was set equal to 1. When robust optimization is faced, the treatment
of these cases is crucial for the optimization process. The quantity of interest to
be minimized, CV aR, depends on the upper tail of the Cumulative Distribution
Function. As a consequence, assigning a high value to the objective in the cases
where convergence is not achieved implies a too high CV aR value that could be
detrimental for the optimization algorithm behaviour. Hence, a proper value of the
objective in these cases must be decided. In particular, in this optimization problem,
the worst objective value selected between the properly converged cases, is assigned
to those in which the error flag is set. Numerical tests lead to conclude that this was
the setup with lowest impact on the optimization process behaviour.
13.5.1 Results
The obtained results are here commented and compared with the baseline airfoil
and the equivalent deterministic solution of the optimization problem. Firstly, in
Fig. 13.3, the airfoil shape of the robust optimized airfoil (dotted line) is compared
with the baseline airfoil (solid line) and the deterministic optimized airfoil (dashed
line).
In addition, the Cumulative Distribution Function obtained by introducing
uncertainties in the airfoil shape are reported in Fig. 13.4 for the initial NACA 2412
airfoil (solid line), for the deterministic optimized airfoil (dashed line), and for the
robust optimized airfoil (dotted line).
The comparison of the CDFs related to the deterministic and robust optimized
airfoils highlights that the robust optimal solution is less vulnerable to uncertainties
in geometric shape with respect to the deterministic one. This can be also deduced