Research on the Karhunen–Loève Transform Method and Its Application to Hull Form Optimization

Abstract

Hull form optimization becomes prone to the curse of dimensionality as the number of design variables increases. The traditional sensitivity analysis method requires massive computational fluid dynamics (CFD) computations and analyzing the effects of all variables on the output; thus, it is extremely time-consuming. Considering this, the development of a rapid and effective dimensionality reduction method is particularly important. The Karhunen–Loève (K–L) transform method projects data from a high-dimensional space onto a low-dimensional space in the direction of the eigenvectors corresponding to large-variance eigenvalues. It extracts the principal components that represent the hull offset information to represent the hull geometric characteristics by analyzing the relationship between the variables in the sample offset matrix. The geometric information matrices of new hull forms can be rapidly reconstructed from the principal components. Compared with direct optimization methods, fewer variables are used to control the deformation of the hull form from the perspective of geometric deformation, avoid a large number of CFD calculations, and improve the efficiency of optimization. This study examined the relevant K–L matrix solution methods and the corresponding hull form reconstruction methods and proposed eigenvalue-based hull form reconstruction equations. The K–L transform method was combined with a previously developed multidisciplinary platform for a comprehensive optimization of ship hydrodynamic performance for hull form optimization, and its effectiveness was verified by using it to optimize DTMB 5415. The results showed that the K–L transform–based dimensionality reduction method significantly reduces the time consumption of optimization while maintaining an acceptable optimization performance.