A Numerical Method for Eigensolution of Near-Regular Structural and Mechanical Systems

Abstract

In this paper a numerical method is developed to find the eigenvalues of the Laplacian matrix for near-regular graph models. Considering the similarity between the pattern of the Laplacian matrix of a graph and the stiffness matrix of a structure, the method can be used for the eigensolution of structural/mechanical systems. Previously, using graph product rules algorithms for the swift solution of the equation F = K∆ in structural/mechanical systems were developed. In this study, using a purposeful ordering along with partitioning the Laplacian/stiffness matrix, a decomposable pattern is achieved. The decomposed parts are solved using combined graph product rules and numerical solutions. While the eigensolution of regular patterns has been traditionally considered as an approximation for near-regular patterns, here the solution of the regular pattern is utilized as an appropriate initial starting point for the solution of near-regular pattern. Furthermore, solving the decomposed regular part using graph product rules reduces the computational complexity of the method.