CFD-based shape optimization under uncertainties using the Adjoint-assisted Polynomial Chaos Expansion and projected derivatives

This paper is dealing with gradient-based optimization in fluid mechanics, in the presence of uncertainties, by focusing on turbulent flows governed by the RANS equations. Uncertainty Quantification (UQ) is performed using the First-Order Second-Moment (FOSM) method and the more accurate Adjoint-assisted (non-intrusive) Polynomial Chaos Expansion (APCE); in the latter, first-order derivatives of the Quantity of Interest (QoI) with respect to the uncertain variables, provided by the adjoint method, are used to reduce the cost for computing the polynomial coefficients. Both methods compute the first two statistical moments (mean value and standard deviation of the QoI) and the objective function to be optimized is their weighted sum. Gradient -based optimization with such an objective function requires mixed derivatives of the QoI with respect to the design and uncertain variables. In adjoint-based optimization, one way to compute these derivatives is by solving the systems of PDEs resulting from the differentiation of the flow and adjoint equations with respect to the uncertain variables, at a cost that scales with their number. To reduce the CPU cost, the objective function gradient can be expressed in terms of the projection of the mixed derivatives' matrix onto vectors. Such projections are herein utilized with both the aforementioned UQ methods, in the context of robust design optimization. Firstly, the projected FOSM method, presented for laminar flows in a previous article by the same group, is extended to turbulent flows solved using the Spalart-Allmaras model by including its adjoint. Then, the projected APCE method is developed, programmed and tested for the first time in the literature. In both pFOSM and pAPCE, the cost for computing the projected matrix of mixed derivatives does not scale with the number of either the design or the uncertain variables. Both methods are implemented within the adjointOptimisation library of OpenFOAM, which makes use of the continuous adjoint method, and are demonstrated in the aerodynamic shape optimization of airfoils, in laminar and turbulent flows.