Dimensionality Reduction Using Principal Component Analysis Applied to the Gradient

Abstract

The cost of dimensionality reduction in aerodynamic design applications involving high-dimensional design spaces and computational fluid dynamics is often prohibitive. In an attempt to overcome this challenge, a new method for dimensionality reduction is presented that scales as , where is the number of design variables. It works by taking advantage of adjoint design methods to compute the covariance matrix of the gradient. This information is then used with principal component analysis to develop a linear transformation that allows an aerodynamic optimization problem to be reformulated in an equivalent coordinate system of lower dimensionality. To demonstrate its feasibility, the method is tested on a two-dimensional staggered airfoil problem, intentionally chosen as an abstraction of a more realistic overwing nacelle integration problem, and shown to exhibit similarities with the latter. Results show that the method rivals typical screening methods used in aerospace engineering in terms of effectiveness but outperforms them either in terms of cost or the ability to capture nonlinear effects. Furthermore, the method is found to show good agreement with optimization results obtained without using the method. Overall, results offer strong evidence in support of the proposed approach, setting the stage for larger analytical efforts such as design space exploration, where dimensionality reduction is unavoidable to cope with the necessity of gradient-free approaches.