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Guaranteed Lower Bounds for the Elastic Eigenvalues by Using the Nonconforming Crouzeix–Raviart Finite Element

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  • Xuqing Zhang

    (School of Mathematical Science, Guizhou Normal University, Guiyang 550001, China
    School of Biology & Engineering, Guizhou Medical University, Guiyang 550025, China)

  • Yu Zhang

    (School of Mathematical Science, Guizhou Normal University, Guiyang 550001, China
    School of Mathematics & Statistics, Guizhou University of Finance and Economics, Guizhou Normal University, Guiyang 550001, China)

  • Yidu Yang

    (School of Mathematical Science, Guizhou Normal University, Guiyang 550001, China)

Abstract

This paper uses a locking-free nonconforming Crouzeix–Raviart finite element to solve the planar linear elastic eigenvalue problem with homogeneous pure displacement boundary condition. Making full use of the Poincaré inequality, we obtain the guaranteed lower bounds for eigenvalues. Besides, we also use the nonconforming Crouzeix–Raviart finite element to the planar linear elastic eigenvalue problem with the pure traction boundary condition, and obtain the guaranteed lower eigenvalue bounds. Finally, numerical experiments with MATLAB program are reported.

Suggested Citation

  • Xuqing Zhang & Yu Zhang & Yidu Yang, 2020. "Guaranteed Lower Bounds for the Elastic Eigenvalues by Using the Nonconforming Crouzeix–Raviart Finite Element," Mathematics, MDPI, vol. 8(8), pages 1-23, July.
    • Handle: RePEc:gam:jmathe:v:8:y:2020:i:8:p:1252-:d:392707
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    References listed on IDEAS

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    1. Liu, Xuefeng, 2015. "A framework of verified eigenvalue bounds for self-adjoint differential operators," Applied Mathematics and Computation, Elsevier, vol. 267(C), pages 341-355.
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