Dimensionality Reduction In Aerodynamic Design Using Principal Component Analysis With Gradient Information

Abstract

The cost of dimensionality reduction in aerodynamic design applications involving highdimensional design spaces and CFD is often prohibitive. In an attempt to overcome this challenge, a new method for dimensionality reduction is presented that scales as p log(p), where p is the number of design variables. It works by taking advantage of adjoint design methods in order to collect gradient observations, which are then used to compute the covariance matrix of the design variables. Dimensionality is then reduced by using principal component analysis to develop a linear transformation that allows an aerodynamic optimization problem to be re-formulated in a new coordinate system of lower dimensionality. In order to demonstrate it's feasibility, the method is tested on a 2D staggered airfoil problem, intentionally chosen as an abstraction of a more realistic over-wing nacelle integration problem, and shown to exhibit similarities with the latter. Results show that the method outperforms typical screening methods used in aerospace engineering, either in terms of effectiveness, cost or the ability to capture nonlinear effects. Furthermore, the method is found to improve convergence during gradient-based optimization. Overall, results offer strong evidence in support of the proposed method, paving the way for larger analytical efforts such as design space exploration, where dimensionality reduction is unavoidable to cope with the necessity of gradient-free approaches.

Aerodynamic Design Optimization and Shape Exploration using Generative Adversarial Networks

“…Aerodynamic design is, in large part, an optimization problem. One common objective is to find design variables that minimize the drag coefficient C D , while maximizing or constraining the lift coefficient C L [9,11,13]. There are primarily three approaches for solving the optimization problem: evolutionary algorithms (EA), surrogate-based optimization (SBO), and automatic differentiation (AD) (otherwise known as adjoint methods).…”

“…In each iteration, the sampling method samples a point in the design space for evaluation of the QoI, and then that point and its QoI update the surrogate model. Compared to methods like genetic algorithms, surrogate-based optimization reduces the number of expensive CFD evaluations needed in aerodynamic shape optimization [7,9,12,14,29]. However, for a high-dimensional design space, the number of evaluations will still be inevitably high due to the curse of dimensionality [6,30].…”

“…These methods fail to consider the correlation between design variables. Thus, researchers have found ways to capture the low-dimensional subspace that identifies important directions with respect to the change of response (i.e., QoI or performance measure) [7][8][9][10][11]. This response-based dimensionality reduction usually has several issues: 1) it requires many simulations when collecting samples of response gradients; 2) variation in gradients can only capture non-linearity rather than variability in the response, so extra heuristics are required to select latent dimensions that capture steep linear response changes; 3) the learned latent space is not reusable for any different design space exploration or optimization task (i.e., when a different response is used); and 4) the linear DR techniques applied in previous work may fail on responses with non-linear correlation between partial derivatives.…”

“…Previous research has looked into dimensionality reduction (DR) of the original parametric design space (i.e., the space of designs represented by shape parameters such as B-spline control points). This permits faster exploration by capturing only those dimensions that either affect the final design's performance [7][8][9][10][11] or capture major shape variability [12][13][14][15][16]. But these DR models may not accurately capture the true variation that we observed in real-world airfoils, e.g., those in the UIUC airfoils database.…”

The Objective Space and the Formulation of Design Requirement in Natural Laminar Flow Optimization

“…The solution proposed in this paper is done via Principal Component Analysis (PCA). PCA is used to strengthen preprocessing of data to improve manipulation efficiency [21,22], dimension reduction [23], and in the sample selection of initialization [24], etc. The methods for data manipulation have been utilized in aerodynamic designs, but mostly in design space than in objective space.…”