Iterative least-squares calculation for modal eigenvector sensitivity

Abstract

There are a number of applications where the sensitivity of eigenvectors with respect to physical parameters is desired. We develop an iterative solution scheme for calculating the eigenvector sensitivity in which only the lowest eigencharacteristics are required. It uses a least-squares formulation for the eigenvector sensitivity including the relation from the basic eigenvalue problem and the orthogonality and normality conditions with respect to the mass matrix. The iterative scheme uses the band structure of the stiffness matrix and an efficient use of the Householder transformation to reduce the number of calculations. Since only the lowest eigensolutions are used in the formulation, it is applicable to situations where only a partial eigensolution of the lowest eigenvectors and eigenvalues is available. The eigenvalues are assumed to be distinct and only the first-order variation is calculated. The stiffness matrix is assumed to be nonsingular and a Choleski decomposition of the stiffness matrix is required, but this is the only large matrix that needs to be decomposed. The least-squares solution to the eigenvector sensitivity is shown to reduce to the modal expansion method when appropriate weights are incorporated. From this expression, we show why the modal expansion is not always adequate for eigenvector sensitivity and give a criterion for evaluating this method in a given application.