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 flow field with a parameterized geometry is a popular test problem for design methodologies. We examine two particular airfoil shape parameterizations, PARSEC and CST, and study the active subspaces present in two common design quantities of interest, transonic lift and drag coefficients, under each shape parameterization. We mathematically relate the two parameterizations with a common polynomial series. The active subspaces enable low-dimensional approximations of lift and drag that relate to physical airfoil properties.In particular, we obtain and interpret a two-dimensional approximation of both transonic lift and drag, and we show how these approximation inform a multi-objective design problem.

G2Aero: A Python package for separable shape tensors

“…The CST method (and other explicit basis representations) often couples linear scaling of the shape (affine deformations) and undulating perturbations. Affine deformations-like changes in thickness, camber, and orientation-are often constrained in design problems (e.g., changes in thickness, Reynolds number, or angle-of-attack) and result in relatively well-understood physical impacts on aerodynamic performance; while undulating perturbations are of increasing interest to airfoil design (Berguin et al, 2015;Glaws, Hokanson, et al, 2022;Grey & Constantine, 2018). G2Aero decouples linear scaling and undulations by defining undulations as the set of all deformations modulo linear scaling of discrete curves.…”

“…Recent rapid development of artificial intelligence (AI) and machine learning (ML) algorithms made an airfoil design a growing area of research once again. Shape representations that better regularize deformations and reduce the dimension of the design space can have a significant impact in AI and ML applications (Chen et al, 2020;Glaws, Hokanson, et al, 2022;Grey & Constantine, 2018).…”

Dimension Reduction via Gaussian Ridge Functions

“…Gaussian ridge functions (posterior mean). Succinctly stated, a Gaussian ridge function is the posterior mean of the Gaussian process in (21), written as (24) ḡ (u) = k u, Û β.…”

“…Across both paradigms, techniques and ideas within subspace-based dimension reduction have proven to be extremely fruitful in inference. Application areas that have benefitted from subspace-based dimension reduction range from airfoil aerodynamics [21] and combustion [28] to economic forecasting (see page 4403 in [2]), turbomachinery aerothermodynamics [35], solar energy [10] and epidemiology [15].…”