Improved version of Gram-Schmidt method for Modal analysis of structures

This paper presents an innovative eigen-solution technique that leverages the Gram-Schmidt procedure to enhance the efficiency of QR-based methods. Among iterative techniques such as vector iteration, polynomial iteration, and subspace iteration, transformation methods have demonstrated superior performance. In particular, the Jacobi and Householder QR methods are well-established in finite element analysis due to their high computational efficiency. Although the Householder QR method is generally preferred over the Jacobi approach for large systems-owing to its initial reduction of the matrix to tridiagonal form, which facilitates subsequent QR iterations-it suffers from a critical drawback: during the reduction process, the bandwidth of the unreduced matrix increases. This expansion disrupts the efficient exploitation of the original banded structure of matrix K. To overcome this limitation, we propose an enhanced version of Gram-Schmidt procedure. This approach supports matrices of various dimensions and provides immediate orthogonal projection, making it superior to conventional methods in computing eigenvalues and eigenvectors. Comparative studies demonstrate that our method yields eigen solutions of comparable accuracy while significantly reducing computation time. Numerical experiments conducted on two types of testing approaches, namely 'two Structural examples, and 'one random-symmetric ill-conditioned matrix. confirm that the proposed technique is both efficient and robust.

کلیدواژه ها:

Eigenproblems ، Transformation methods ، Householder-QR iteration ، Gram-Schmidt