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Optimal modal reduction of dynamic subsystems: Extensions and improvements

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Abstract

The problem of optimal reduction of a linear dynamic subsystem is revisited. The subsystem may represent, for example, a complex but minor part of a large elastic structure. The goal is to drastically reduce the number of degrees of freedom of a subsystem attached through an interface to a main system in such a way as to affect the dynamic behavior of the main system in the least possible way. In a recent publication, the Optimal Modal Reduction (OMR) algorithm was developed to this end. This algorithm seeks a reduction of the subsystem that will have minimal effect, in the L2 norm, on the Dirichlet-to-Neumann (DtN) map on the interface. Here this algorithm is extended and improved in a number of ways. First, a family of alternative formulations are derived for the DtN map, which lead to alternative OMR algorithms; one of them, called OMR j=2, is shown to yield better results than the original formulation. Second, the OMR algorithm, which was originally developed for undamped subsystems, is extended to subsystems undergoing Rayleigh damping. In addition, an enhanced derivation of the original OMR algorithm is presented, and the good pointwise performance of OMR is explained by relating to minimization with respect to the H1 norm. The extensions mentioned above are discussed theoretically as well as demonstrated via numerical examples. Experiments and discussion include comparison of the OMR algorithm to simple coarsening of the subsystem discretization. In all cases, central finite difference discretization in space and explicit time-stepping are employed to solve the scalar wave equation. Copyright © 2010 John Wiley & Sons, Ltd.

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