NLEVP: A Collection of Nonlinear Eigenvalue Problems
ACM Transactions on Mathematical Software (TOMS)
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Numerical simulation enables the study of complex physical problems for which hand analysis is intractable and experiments are difficult, expensive, or dangerous. Algorithm design and analysis are at the heart of numerical simulation. This dissertation consists of two parts under a unified theme: the design and analysis of structure-preserving subspace projection algorithms for model order reduction and nonlinear eigenvalue computation. In the first part, we study substructuring methods for model order reduction. These methods include the component mode synthesis (CMS) method, which has been studied since the 1960s. The modes of subsystems associated with the lowest frequencies are typically retained in these methods. This is largely heuristic and does not necessarily produce an optimal reduced system. We present a coupling-matrix based mode selection scheme for the CMS method, referred to as the CMSχ method. This new scheme is derived using a moment-matching principle defined on the interface between substructures. It is compatible to the one in recently proposed optimal modal reduction (OMR) method. The improvements of the CMSχ method to the CMS and OMR methods are demonstrated by numerical examples from structural dynamics in both the frequency and time domain. In the second part, we study numerical methods for solving large-scale nonlinear eigenvalue problems arising from finite element analysis of external Q of waveguide-loaded resonant cavities in the design of next-generation particle accelerators. We present a structure-preserving nonlinear Rayleigh-Ritz iterative subspace projection algorithm, NRRIT in short, and demonstrate that NRRIT is a promising approach for solving large scale nonlinear eigenvalue problems in two cavity designs.