Block-partitioned Rayleigh-Ritz method for efficient eigenpair reanalysis of large-scale finite element models

Abstract

In this manuscript, we propose a new effective method for eigenpair reanalysis of large-scale finite element (FE) models. Our method utilizes the matrix block-partitioning algorithm in the Rayleigh-Ritz approach and expresses the Ritz basis matrix using thousands of block matrices of very small size. To avoid significant computational costs from the projection procedure, we derive a new formulation that uses tiny block computations instead of global matrix computations. Additionally, we present an algorithm that recognizes which blocks are changed in the modified FE model to achieve computational cost savings when computing new eigenpairs. Through selective updating for the recognized blocks, we can effectively construct the new Ritz basis matrix and the new reduced mass and stiffness matrices corresponding to the modified FE model. To demonstrate the performance of our proposed method, we solve several practical engineering problems and compare the results with those of the combined approximation (CA) method, the most well-known eigenpair reanalysis method, and ARPACK, an eigenvalue solver embedded in many numerical programs.