Adjoint-based robust optimization using polynomial chaos expansions

Abstract

Adjoint methods are nowadays widely used to efficiently perform optimization for problems with a large number of design variables. However, in reality, the problem at hand might be subjected to uncertainties in the operational conditions or, in case of optimizing geometries, the design variables itself might be uncertain due to manufacturing tolerances. For such applications, the optimum obtained using deterministic methods might be very sensitive to small variations in the uncertainties, i.e. it lacks robustness. In a robust optimization, the uncertainties are taken directly into account during the optimization process by introducing, next to the mean objective, its variance as a second objective. This implies that, when using gradient based optimization methods, the gradients of both objectives (mean and variance) must be known. In this work the Polynomial Chaos Expansion (PCE) is used in combination with adjoint methods to efficiently obtain both gradients. A non intrusive, regression based PCE is used, requiring a new adjoint solution for each sampling point in order to build the PCE of the gradient. A PCE for the objective is also built (at no extra cost) in order to compute the gradient of the variance. A weighted average of both gradients is then used to find an optimum. By changing the weighting factors the solution can be found favouring either of the two objectives. The developed approach is applied to relevant engineering problems, such as geometrical optimization of pipe flows and flow over airfoils. The design variables are the shape coordinates and no parameterization is used. In this work, the design variables were considered deterministic with the uncertainties coming from operational conditions.