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Solving large-scale finite element nonlinear eigenvalue problems by resolvent sampling based Rayleigh-Ritz method

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Abstract

This paper focuses on the development and engineering applications of a new resolvent sampling based Rayleigh-Ritz method (RSRR) for solving large-scale nonlinear eigenvalue problems (NEPs) in finite element analysis. There are three contributions. First, to generate reliable eigenspaces the resolvent sampling scheme is derived from Keldysh’s theorem for holomorphic matrix functions following a more concise and insightful algebraic framework. Second, based on the new derivation a two-stage solution strategy is proposed for solving large-scale NEPs, which can greatly enhance the computational cost and accuracy of the RSRR. The effects of the user-defined parameters are studied, which provides a useful guide for real applications. Finally, the RSRR and the two-stage scheme is applied to solve two NEPs in the FE analysis of viscoelastic damping structures with up to 1 million degrees of freedom. The method is versatile, robust and suitable for parallelization, and can be easily implemented into other packages.

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Acknowledgements

JX gratefully acknowledges the financial supports from the National Science Foundations of China under Grants 11102154 and 11472217, Fundamental Research Funds for the Central Universities in China, and the Alexander von Humboldt Foundation (AvH) to support his research fellowship at the Chair of Structural Mechanics, University of Siegen, Germany.

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Authors and Affiliations

  1. School of Astronautics, Northwestern Polytechnical University, Xi’an, 710072, China

    Jinyou Xiao, Hang Zhou & Chao Xu

  2. Department of Civil Engineering, University of Siegen, 57068, Siegen, Germany

    Chuanzeng Zhang

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  1. Jinyou Xiao
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  2. Hang Zhou
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  3. Chuanzeng Zhang
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  4. Chao Xu
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Correspondence to Jinyou Xiao.

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Xiao, J., Zhou, H., Zhang, C. et al. Solving large-scale finite element nonlinear eigenvalue problems by resolvent sampling based Rayleigh-Ritz method. Comput Mech 59, 317–334 (2017). https://doi.org/10.1007/s00466-016-1353-4

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  • DOI: https://doi.org/10.1007/s00466-016-1353-4

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