Mathematical Description of the Uncertainties

Since we want to avoid a parametrization of the uncertainties which would lead to a reduction of the space of realizations, we choose a stochastic approach in or­der to include the uncertainties in the optimization problem. Furthermore, this ap­proach allows to adapt the robust optimization to new information of the uncertain parameter, e. g. if new measurements are available, so that a general framework of robust aerodynamic design can be developed.

The proper treatment of the uncertainties within a numerical context is a very im­portant challenge, since the simulation and also optimization under uncertainties is a fast growing field of research. Again, we distinguish between the uncertainties with respect to the flight conditions, the scalar-valued uncertainties, and the geometrical uncertainties, the function-valued uncertainties.

1.1.1 Scalar-Valued Uncertainties

The scalar-valued uncertainties, e. g. the Mach number, are modeled as real-valued, continuous random variables

у: Q ^ R, (1)

defined on a given probability space (Q, Y,P). They are characterized by a given probability density function

Фtruncated : R ^ R+ • (2)

We assume (mainly due to lack of statistical data) a truncated normal distribution of the perturbations ensuring that the realizations lie in between the given bounds. Furthermore, the mean value of the random variable corresponds with the value of the deterministic model. These assumptions are widely used in order to describe uncertainties in CFD (cf. [32]). Nevertheless, the model need to be adapted to meas­urements, if available.