Karhunen-Loève expansion for assessing stochastic subspaces in geometry optimization and geometric uncertainty quantification

The Karhunen-Loève Expansion (KLE) is a method for identifying the principal orthogonal directions in a stochastic process or space. Here, the expansion is applied to assess a geometry-optimization research space before going through the minimization process. The research space is defined by blending a set of initial geometries through a morphing technique. The KLE gives the total geometric variability related to the research space, and provides with the geometric variability associated to each principal direction. This allows for assessing the the research space prior going through the optimization process. Moreover, the KLE allows for a rational reduction of the research space dimension (often required to achieve better convergence of the optimization algorithm and to reduce the computational effort) and for the identification of the relevant variables for both geometry-optimization and geometric uncertainty quantification (UQ). The aim of this report is to provide with the formulation for the KLE in the context of geometry optimization. Also, the application of the method to the Delft catamaran bare- hull optimization problem is addressed.