Abstract:
This paper proposes a novel multi-level substructuring method based on Proper Orthogonal Decomposition (POD). The method introduces the concept of master-slave degrees of freedom and extends traditional static condensation through further order reduction. Specifically, modal decomposition and Singular Value Decomposition (SVD) techniques are employed to construct low-order modal bases and high-order approximate modal bases. These bases are utilized to reduce the numerical basis functions in static condensation and approximate the internal stiffness matrix, significantly decreasing storage requirements during computation. To achieve efficient connectivity between substructures, orthogonal bases capable of representing all substructural deformations within a unified linear space are derived by applying SVD to the low-order modal components of substructural boundaries. These orthogonal bases enable additional dimensionality reduction of substructural interface matrices, substantially improving computational speed in multi-level substructuring. Furthermore, the paper elaborates on substructural assembly techniques and presents an innovative approach to address the issue of rigid body modes corresponding to zero eigenvalues. Quantitative analysis of temporal and spatial complexity demonstrates that the proposed method not only achieves remarkable improvements in spatial complexity but also effectively controls computational costs for eigenvalue problems in vibration mode analysis. Numerical examples validate the method's effectiveness and reliability, showing enhanced numerical accuracy, convergence performance, and computational efficiency with increasing numbers of orthogonal bases.