Development and assessment of an intrusive polynomial chaos expansion‐based continuous adjoint method for shape optimization under uncertainties

This article contributes to the development of methods for shape optimization under uncertainties, associated with the flow conditions, based on intrusive Polynomial Chaos Expansion (iPCE) and continuous adjoint. The iPCE to the Navier–Stokes equations for laminar flows of incompressible fluids is developed to compute statistical moments of the Quantity of Interest which are, then, compared with those obtained through the Monte Carlo method. The optimization is carried out using a continuous adjoint‐enabled, gradient‐based loop. Two different formulations for the continuous adjoint to the iPCE PDEs are derived, programmed, and verified. Intrusive PCE methods for the computation of the statistical moments require mathematical development, derivation of a new system of governing equations and their numerical solution. The development is presented for a chaos order of two and two uncertain variables and can be used as a guide to those willing to extend this development to a different set of uncertain variables or chaos order. The developed method and software, programmed in OpenFOAM, is applied to two optimization problems pertaining to the flow around isolated airfoils with uncertain farfield conditions. The intrusive Polynomial Chaos Expansion (iPCE) to the Navier–Stokes equations for laminar and incompressible flows is mathematically developed, along with its continuous adjoint counterpart, to support aerodynamic shape optimization under uncertainties. Two variants of the adjoint equations are developed, which prove to be analytically and numerically equivalent. The primal and adjoint iPCE solvers, developed in OpenFOAM, are validated against Monte Carlo and finite differences, respectively, and are, then, used for the shape optimization of two airfoils with uncertain farfield conditions.