pFOSM: an efficient algorithm for aerodynamic robust design based on continuous adjoint and matrix-vector products. This article proposes a novel robust design tool based on the method of moments (MoM), demonstrated in problems governed by laminar flows of incompressible fluids. The objective function of the robust design problem consists of the weighted sum of the mean value and standard deviation of the quantity of interest (QoI), computed based on the value and gradient of the latter with respect to the uncertain variables. First-order derivatives with respect to the uncertain and design variables are computed using the continuous adjoint method. Using a first-order MoM to compute the objective function and by minimizing it with a gradient-based approach leads to the need of computing up to second-order derivatives of the QoI with respect to the design and uncertain variables. This can lead to a high computational cost in problems with a great number of uncertain variables. The proposed robust design algorithm avoids computing second-order derivatives by computing matrix-vector products instead, using a combination of adjoint and direct differentiation. This makes the cost per gradient computation independent of the numbers of both the design and uncertain variables. The proposed method is demonstrated through the constrained shape optimization of an isolated 2D airfoil under uncertain farfield flow conditions.

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