Design-space dimensionality reduction for single-and multi-disciplinary shape optimization

The present paper presents an offline dimensionality reduction technique for single- and multi-disciplinary shape optimization problems, based on a generalized Karhunen-Loève expansion (KLE) of the shape modification vector. The formulation is based on the geometric variability and the method does not require objective function or gradient evaluations. The mathematical derivation of the design-space dimensionality reduction is presented for a global representation of the shape modification. The associated structure and breakdown of the geometric variance is investigated through eigenvalues and eigenmodes provided by the KLE. The dimensionality reduction for the shape optimization problem is based on the eigenvalues, which represent the geometric variance associated to the corresponding eigenmodes. The reduced-dimensionality design space is defined using the eigenmodes as new basis functions. Two example applications are presented. The first example is the hydrodynamic hull-form optimization for resistance reduction of an USS Arleigh Burke-class destroyer ship. The results show the KLE capability of reducing the design-space dimensionality, while retaining a prescribed level of design variability and achieving the same optimization results as the original design space. The second example is the design-space dimensionality reduction of a NACA 0009 3D hydrofoil, used as multi-disciplinary design optimization test case in ongoing research by the authors. The formulation presented goes beyond the current applications and is suitable in all areas where shape design is of primary importance (such as aerodynamics, aeroelasticity, structural, and heat transfer applications), involving complex single and multi-disciplinary simulations.