Aerodynamic shape optimization for minimum robust drag and lift reliability constraint

A methodology for shape optimization of aerodynamic bodies under uncertainties is presented. Flow-related and geometrical uncertainties are considered and quantified by probability distribution functions. The optimal shape is computed by minimizing a robust estimate of the drag coefficient subject to reliability constraint for the lift coefficient. The robust drag is formulated as a weighted sum of the mean and the standard deviation of the drag coefficient over the space of uncertain parameters. The mean and standard deviation of the drag coefficient are computed using sparse grid techniques. The lift reliability, defined by the probability the lift coefficient is lower than a reference value, is computed using First Order Reliability Method (FORM). A gradient-based optimization algorithm is used to obtain the optimal shape. The sensitivity derivatives of robust drag measure and the lift reliability with respect to the shape controlling and flow related design parameters as well as the uncertain parameters are computed using the adjoint problem for the flow. The methodology is applied to pure aerodynamic shape optimization, comparing optimal designs that arise from the formulation to optimal designs that correspond to special cases, including the case of no uncertainties. A 2D airfoil case is designed based on the Euler equations under uncertain Mach number and angle of attack and geometric variability.