| Models of complex engineering structures, such as automotive assemblies, often contain much more information than is necessary to understand the vibratory characteristics that influence design and testing. Condensation methods such as modal condensation and Krylov projection methods extract such characteristics directly from the matrix equations of motion, and are intimately related to iterative methods of matrix inversion. This research focuses on two main points: error bounds to evaluate the quality of a given approximation; the introduction of the notion of "Generalized Krylov subspaces" to improve the quality of approximations.; An important contribution of the present work is the development of error bounds that can be efficiently evaluated. Although error estimates exist for some condensation methods, generally applicable error bounds are lacking in the literature. We present methodologies to derive such error bounds for any linear combination of the components of the solution. We derive several distinct bounds following these methodologies. The bounds are based on new expressions of the error and on error bounds for system eigenpairs. For this purpose, new error bounds on system eigenpairs have been developed, which are much tighter than previously known bounds. The error bounds can be specialized and computed cheaply and robustly for many condensation methods. In particular, for all Krylov projection methods, methods are derived to update the bounds cheaply from one iteration to the next or to compute them implicitly.; The second major contribution of this work is the introduction and characterization of "Generalized Krylov subspaces." These have the characteristic of a dimension that grows exponentially with the number of iterations. The use of Generalized Krylov subspaces as defined herein conceptually unifies many heretofore apparently disparate concepts, including block Krylov methods, rational Krylov methods, and inexact Krylov methods. Work in this area is still in its infancy. Here we have succeeded in identifying several canonical bases and canonical functions to be used in iterative methods in generalized Krylov subspaces. |
