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An h-version adaptive FEM for eigenproblems in system of second order ODEs: vector Sturm-Liouville problems and free vibration of curved beams

Yongliang Wang (School of Mechanical and Civil Engineering, State Key Laboratory of Coal Resources and Safe Mining, China University of Mining and Technology, Beijing, China)

Engineering Computations

ISSN: 0264-4401

Article publication date: 8 September 2020

Issue publication date: 17 June 2021

190

Abstract

Purpose

This study aims to overcome the involved challenging issues and provide high-precision eigensolutions. General eigenproblems in the system of ordinary differential equations (ODEs) serve as mathematical models for vector Sturm-Liouville (SL) and free vibration problems. High-precision eigenvalue and eigenfunction solutions are crucial bases for the reliable dynamic analysis of structures. However, solutions that meet the error tolerances specified are difficult to obtain for issues such as coefficients of variable matrices, coincident and adjacent approximate eigenvalues, continuous orders of eigenpairs and varying boundary conditions.

Design/methodology/approach

This study presents an h-version adaptive finite element method based on the superconvergent patch recovery displacement method for eigenproblems in system of second-order ODEs. The high-order shape function interpolation technique is further introduced to acquire superconvergent solution of eigenfunction, and superconvergent solution of eigenvalue is obtained by computing the Rayleigh quotient. Superconvergent solution of eigenfunction is used to estimate the error of finite element solution in the energy norm. The mesh is then, subdivided to generate an improved mesh, based on the error.

Findings

Representative eigenproblems examples, containing typical vector SL and free vibration of beams problems involved the aforementioned challenging issues, are selected to evaluate the accuracy and reliability of the proposed method. Non-uniform refined meshes are established to suit eigenfunctions change, and numerical solutions satisfy the pre-specified error tolerance.

Originality/value

The proposed combination of methodologies described in the paper, leads to a powerful h-version mesh refinement algorithm for eigenproblems in system of second-order ODEs, that can be extended to other classes of applications in damage detection of multiple cracks in structures based on the high-precision eigensolutions.

Keywords

Acknowledgements

This work was supported by the National Natural Science Foundation of China (grants 41877275 and 51608301), Yue Qi Young Scholar Project Foundation of China University of Mining and Technology, Beijing (grant 2019QN14), Fundamental Research Funds for the Central Universities, Ministry of Education of China (grant 2019QL02), Teaching Reform and Research Projects of Undergraduate Education of China University of Mining and Technology, Beijing (grant J200709 and J190701) and the Open Fund of Tianjin Key Lab of Soft Soil Characteristics and Engineering Environment (grant 2017SCEEKL003).

Citation

Wang, Y. (2021), "An h-version adaptive FEM for eigenproblems in system of second order ODEs: vector Sturm-Liouville problems and free vibration of curved beams", Engineering Computations, Vol. 38 No. 4, pp. 1807-1830. https://doi.org/10.1108/EC-05-2020-0242

Publisher

:

Emerald Publishing Limited

Copyright © 2020, Emerald Publishing Limited

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