Optimization Under Mixed Aleatory/Epistemic Uncertainty Using Derivatives

Abstract

In this paper mixed aleatory/epistemic uncertainties in a robust design problem are propagated via the use of box-constrained optimizations and surrogate models. The assumption is that the uncertain input parameters can be divided into a set only containing aleatory uncertainties and a set with only epistemic uncertainties. Uncertainties due to the epistemic inputs can then be propagated via a box-constrained optimization approach, while the uncertainties due to aleatory inputs can be propagated via sampling. A statisticsof-intervals approach is used in which the box-constrained optimization results are treated as a random variable and multiple optimizations need to be performed to quantify the aleatory uncertainties via sampling. A Kriging surrogate is employed to model the variation of the optimization results with respect to the aleatory variables enabling exhaustive Monte-Carlo sampling to determine the desired statistics for each robust design iteration. This approach is applied to the robust design of a transonic NACA 0012 airfoil where shape design variables are assumed to have epistemic uncertainties and the angle of attack and Mach number are considered to have aleatory uncertainties. The very good scalability of the framework in the number of epistemic variables is demonstrated as well. Nomenclature α = Aleatory variables β = Epistemic variables C d = Drag coefficient C l = Lift coefficient D = Design variables f = Simulation output f = Mean of simulation output f = Kriging predictor for simulation output f ext = Represents either max β f or min β f g = Constraints J = Objective function dJ dD = Gradient of objective function M = Number of design variables N = Number of Kriging training points N = Number of sampling points P k = Desired or specified probability q = State variables R = State equation residual s = Epistemic uncertainty bound σ = Standard deviation Var = Variance w i = User specified weights