Matrix Analysis of Repetitive Circulant Structures: New-block and Near Block Matrices

Abstract

In many scientific fields and several problems, Block Circulant Matrices (BCM) have been used for a long period of time. Each row of the BCM is a cyclic shift of its upper row to the right. BCM has been studied widely and there are closed-form solutions for problems of BCM. In these problems, the properties of near-BCM and BCM lead to a significant decrease in computational cost and efforts. In other words, these matrices are useful to perform some computational operations at the low cost. This study introduces a method for transforming a structure into a new type of Block Circulant Structure (BCS) by applying minor modifications. Furthermore, transformation of structural matrices into Block Circulant Matrices is discussed, and the properties of these matrices are then described in details. The methods introduce calculating eigenvalues and eigenvectors of these matrices instead of calculating the inverse of their matrices. To achieve this goal, the properties of near-Block and Block Circulant Matrices are used to analyze the structural stiffness matrices. In addition, the inverse of stiffness matrices for structures are calculated and utilized in structural mechanics. For clarification of efficiency and accuracy of the method, some examples are presented.