Energy preservation in POD based reduced order models for linearly vibrating systems

Hossain, MN and Bharti, C and Ghosh, D (2023) Energy preservation in POD based reduced order models for linearly vibrating systems. In: Mechanics Research Communications, 128 .

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Abstract

The cost of computing the response of a dynamical system grows significantly with the fidelity of the model. Development of a reduced order model (ROM) can alleviate this cost greatly. A proper orthogonal decomposition (POD) based ROM is created by projecting the full-scale model in a lower dimensional subspace obtained from a singular value decomposition of response snapshots. It can be created either in the configuration space or state space. The POD bases have been successful in capturing the maximum displacement and velocity. However, their accuracy in preserving the energy of the vibrating system has not been explored. In this work, this aspect is explored and a new mathematical formulation is proposed that improves the accuracy of the ROM in preserving the energy. Accordingly a new method is proposed where an optimal set of POD bases is found by using preservation of time averaged total mechanical energy – the sum of kinetic and potential energies – in the state space. This energy is of primary concern for undamped or lightly damped systems. The energy being one fundamental quantity in vibration, this new formulation gives POD bases a strong theoretical underpinning. Through detailed numerical studies it is observed that for a given size the proposed method is more accurate than the existing ROM in state space. The improved accuracy is more pronounced for undamped or lightly damped systems. The proposed ROM is accurate even for systems with clustered natural frequencies.

Item Type:	Journal Article
Publication:	Mechanics Research Communications
Publisher:	Elsevier Ltd
Additional Information:	The copyright for this article belongs to Elsevier Ltd.
Keywords:	Dynamical systems; Numerical methods; Potential energy; Principal component analysis, Energy; Energy preservation; Orthogonal decomposition; Proper Orthogonal; Proper orthogonal decomposition; Reduced order modelling; Reduced-order model; Soil-structure interaction; Spatially periodic systems; State-space, Singular value decomposition
Department/Centre:	Division of Mechanical Sciences > Civil Engineering
Date Deposited:	10 Feb 2023 04:21
Last Modified:	10 Feb 2023 04:21
URI:	https://eprints.iisc.ac.in/id/eprint/80138

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