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TRAX: An approach for the time rational analysis of complex dynamic systems
Author(s) -
Lecomte Christophe
Publication year - 2015
Publication title -
international journal for numerical methods in engineering
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.421
H-Index - 168
eISSN - 1097-0207
pISSN - 0029-5981
DOI - 10.1002/nme.4841
Subject(s) - linear subspace , krylov subspace , context (archaeology) , interpolation (computer graphics) , computer science , dynamical systems theory , transfer function , complex system , frequency domain , rational function , linear system , algorithm , matrix (chemical analysis) , frequency response , mathematical optimization , mathematics , iterative method , artificial intelligence , motion (physics) , paleontology , mathematical analysis , physics , geometry , materials science , quantum mechanics , electrical engineering , composite material , computer vision , biology , engineering
Summary This paper introduces a theoretical and algorithmic reduced model approach to efficiently evaluate time responses of complex dynamic systems. The proposed approach combines four main components: analytical expressions of the average of the system's transfer functions in the frequency domain, precise and convergent rational approximations of these exact expressions, exact evaluation of these approximations through model reduction in rational Krylov subspaces and semi‐analytical interpolation at just a few frequency points. The resulting algorithmic principles to evaluate the time response of a particular system are relatively straightforward: one first evaluates the response of the system with slight additional damping at a few frequencies and one then projects or reduces the system in the subspace spanned by these responses. The time response of the reduced model implicitly provides a precise evaluation of that of the original system. The properties of the reduced models and the precision of the proposed approach are studied, and applications on complex matrix systems are presented and discussed. While the theory and numerical algorithms are presented in a matrix context, they are also transposable in a continuous functional context. Copyright © 2015 John Wiley & Sons, Ltd.