A practical algorithm for the efficient computation of eigenvector sensitivities

Derivatives of eigenvalues and eigenvectors have become increasingly important in the development of modern numerical methods for areas such as structural design optimization, dynamic system identification and dynamic control, and the development of effective and efficient methods for the calculation of such derivatives has remained to be an active research area for several decades. In this paper, a practical algorithm has been developed for efficiently computing eigenvector derivatives of generalized symmetric eigenvalue problems. For eigenvector derivative of a separate mode, the computation only requires the knowledge of eigenvalue and eigenvector of the mode itself and an inverse of system matrix accounts for most computation cost involved. In the case of two close modes, the modal information of both modes is required and the eigenvector derivatives can be accurately determined simultaneously at minor additional computational cost. Further, the proposed method has been extended to the case of practical structural design where structural modifications are made locally and the eigenderivatives of the modes concerned before are still of interest. By combining the proposed algorithm together with the proposed inverse iteration technique and singular value decomposition theory, eigenproperties and their derivatives can be very efficiently computed. Numerical results from a practical finite element model have demonstrated the practicality of the proposed method. The proposed method can be easily incorporated into commercial finite element packages to improve the computational efficiency of eigenderivatives needed for practical applications.