Gene Golub Symposium
Inverse Eigenvalue Problems: Gene's All Time Favorites
(slides)
Biswa Nath Datta, Northern Illinois University
Inverse eigenvalue problem for a matrix A concerns constructing the matrix A from a prescribed set of eigenvalues and eigenvectors. Gene Golub made some pioneering contributions to this topic. His work with Dan Boley, Carl de Boor, Moody Chu, and several others, has much enriched this topic. His book on "Inverse Eigenvalue Problems" (co-authored with Moody Chu) is a rich source of knowledge on inverse eigenvalue problems.
My talk today will deal with Finite Element Model Updating(FEMU), which is a special inverse eigenvalue problem for a quadratic matrix pencil. Gene expressed a keen interest in this problem after listening to my talk on this topic last year delivered at the workshop on "Numerical Linear Algebra in Signals, Systems, and Control", held in Indian Institute of Technology-Kharagpur, India.
The FEMUP arises in vibration industries in the context of designing automobiles, air and space crafts,and others. The problem is to update a theoretical finite element model using only a few measured data from a real-life structure which was designed based on the theoretical model. The model has to be updated in such a way that the measured eigenvalues and eigenvectors will be incorporated into the model, the symmetry of the original model will be preserved and the eigenvalues and eigenvectors that do not participate in updating will remain unchanged. When the model has been updated this way, the updated model can be used for future design with confidence. Finite Element Model Updating has also useful applications in health monitoring and damage detection in structures, including bridges, buildings, highways, and others.
Despite much research done on the problem both by academic and industrial researchers and engineers, the problem has not been satisfactorily solved and an active research is still underway. There are many industrial solutions which are ad hoc in nature and often lack solid mathematical foundations. In this talk, I shall present a brief overview of the existing techniques and their practical difficulties, along with new developments that happened within the last few years. The talk will conclude with a few words on future research direction on this topic.
