Geometric Filtration Using Proper Orthogonal Decomposition for Aerodynamic Design Optimization
David J. J. Toal, Neil W. Bressloff, Andy J. Keane, Carren M. E. Holden · AIAA Journal · 2010
When carrying out design searches, traditional variable screening techniques can find it extremely difficult to distinguish between important and unimportant variables. This is particularly true when only a small number of simulations are combined with a parameterization that results in a large number of variables of seemingly equal importance. Here, the authors present a variable reduction technique that employs proper orthogonal decomposition to filter out undesirable or badly performing geometries from an optimization process. Unlike traditional screening techniques, the presented method operates at the geometric level instead of the variable level. The filtering process uses the designs that result from a geometry parameterization instead of the variables that control the parameterization. The method is shown to perform well in the optimization of a two-dimensional airfoil for the minimization of drag-to-lift ratio, producing designs better than those resulting from traditional krigingbased surrogate model optimization and with a significant reduction in surrogate tuning cost. Nomenclature C = fluctuation correlation matrix CD = drag coefficient CL = lift coefficient F = matrix of snapshot fluctuations l = design variable lower bounds M = no. of snapshots n = no. of sample points p = hyperparameter determining smoothness R = correlation matrix S = snapshot ensemble s = snapshot vector u = design variable upper bounds V = matrix of eigenvectors x = design variable y = objective function = modal coefficient = hyperparameter determining correlation = vector of eigenvalues = eigenvalue = mean = standard deviation = matrix of eigenfunctions I.