Second-order sensitivity of eigenpairs in multiple parameter structures

This paper presents methods for computing a second-order sensitivity matrix and the Hessian matrix of eigenvalues and eigenvectors of multiple parameter structures. Second-order perturbations of eigenvalues and eigenvectors are transformed into multiple parameter forms, and the second-order perturbation sensitivity matrices of eigenvalues and eigenvectors are developed. With these formulations, the efficient methods based on the second-order Taylor expansion and second-order perturbation are obtained to estimate changes of eigenvalues and eigenvectors when the design parameters are changed. The presented method avoids direct differential operation, and thus reduces difficulty for computing the second-order sensitivity matrices of eigenpairs. A numerical example is given to demonstrate application and accuracy of the proposed method.