Design-space dimensionality reduction in shape optimization by Karhunen-Loève expansion

The paper presents a methodology to reduce the dimension of design spaces in shape optimization problems, while retaining a desired level of geometric variance. The method is based on a generalized Karhunen-Loève expansion (KLE). Arbitrary shape modification spaces are assessed in terms of Karhunen-Loève modes (eigenvectors) and associated geometric variance (eigenvalues). The former are used as a basis in order to build a reduced-dimensionality representation of the shape modification. The method is demonstrated for the shape optimization of a high-speed catamaran, based on CFD simulations and aimed at the reduction of the wave component of calm-water resistance. KLE is applied to three design spaces with large dimensionality (≥20), based on a free form deformation technique. The space with the largest geometric variance is selected for dimensionality reduction and design optimization. N-dimensional design spaces are used, with N=1, 2, 3, and 4, retaining up to the 95% of the geometric variance associated to the original space. The correlation between the objective reduction achieved, the dimension N and the geometric variance of the reduced-dimensionality space is shown and found significant.