Towards Augmented Design-Space Exploration via Combined Geometry and Physics Based Karhunen-Loève Expansion

The paper presents an offline method to reduce the dimensionality of design spaces in stochastic shape optimization with multiple environmental and operating conditions, offering opportunities for an augmented design-space exploration. A generalized Karhunen-Loève expansion is applied to a combined geometry and physics based design modification vector, evaluated for multiple design conditions and embedded in a gener- alized (disjoint) Hilbert space. Design spaces are assessed in terms of Karhunen-Lo` ve (KL) modes (eigenvectors) and associated variance (eigenvalues). The former are used as a basis to build a reduced-dimensionality representation of the shape modification, retaining the 95% of the original design variability. The dimensionality reduction formulation is independent of the shape parametrization technique used. The effectiveness of the method is demonstrated for a naval hydrodynamic problem, namely the design-space dimensionality reduction of a destroyer-type hull for robust shape optimization addressing the reduction of wave resistance at multiple speeds. The dimensionality reduction is based on geometry, wave resistance coefficient, pressure distribution, and wave elevation, evaluated by linear potential flow theory. The method presented goes beyond the current application and is suitable in all areas where shape design is of primary importance (such as fluid dynamics, structural, and heat transfer applications), involving complex simulations. The extension to combined geometry and physics based analysis, including multiple speeds, allows for an augmented design-space assessment through an extended representation of the KL modes to physics based information for multiple design conditions.