Active Subspaces of Airfoil Shape Parameterizations

Abstract

Design and optimization benefit from understanding the dependence of a quantity of interest (e.g., a design objective or constraint function) on the design variables. A low-dimensional active subspace, when present, identifies important directions in the space of design variables; perturbing a design along the active subspace associated with a particular quantity of interest changes that quantity more, on average, than perturbing the design orthogonally to the active subspace. This low-dimensional structure provides insights that characterize the dependence of quantities of interest on design variables. Airfoil design in a transonic flowfield with a parameterized geometry is a popular test problem for design methodologies. Two particular airfoil shape parameterizations are examined (parametric sections and class-shape transformation), and the active subspaces present in two common design quantities of interest (transonic lift and wave-drag coefficients) are studied under each shape parameterization. The two parameterizations are mathematically related by a simple change of variable and subsequent odd-power polynomial series. The active subspaces enable low-dimensional approximations of lift and drag that relate to physical perturbations of the airfoil. In particular, a two-dimensional approximation of both transonic lift and drag is obtained and interpreted, where 1) each dimension is a particular linear combination of the design variables resulting in intuitive airfoil perturbations, and 2) the linear combination weights are related between lift and drag. It is shown how these approximations along with similarities between the lift and drag active subspaces inform a multi-objective design problem.