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Research Article

Numerical method for two-dimensional linearly elastic clamped plate model

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Pages 1779-1793 | Received 05 Aug 2022, Accepted 09 May 2023, Published online: 29 May 2023

References

  • J. Ahrens, B. Geveci, and C. Law, ParaView: An End-User Tool for Large Data Visualization, Visualization Handbook, Elsevier, 2005.
  • M. Aouadi, Global and exponential attractors for extensible thermoelastic plate with time-varying delay, J. Differ. Equ. 269(5) (2020), pp. 4079–4115.
  • D.N. Arnold and F. Brezzi, Locking-free finite element methods for shells, Math. Comput. 66(217) (1997), pp. 1–15.
  • D.N. Arnold, F. Brezzi, and L.D. Marini, A family of discontinuous Galerkin finite elements for the Reissner–Mindlin plate, J. Sci. Comput. 22–23(1–3) (2005), pp. 25–45.
  • A.K. Aziz, The Mathematical Foundations of the Finite Element Method with Applications to Partial Differential Equations, Academic Press, New York, 1972.
  • S.C. Brenner and L.R. Scott, The Mathematical Theory of Finite Element Methods, Springer, New York, 2008.
  • S.C. Brenner, L.Y. Sung, and Y. Zhang, Finite element methods for the displacement obstacle problem of clamped plates, Math. Comput. 81(279) (2012), pp. 1247–1262.
  • S.C. Brenner, L.Y. Sung, H.C. Zhang, and Y. Zhang, A Morley finite element method for the displacement obstacle problem of clamped Kirchhoff plates, J. Comput. Appl. Math. 254(1) (2013), pp. 31–42.
  • H.R. Chen, S.C. Chen, and Z.H. Qiao, C0-Nonconforming triangular prism elements for the three-dimensional fourth order elliptic problem, J. Sci. Comput. 55(3) (2013), pp. 645–658.
  • P.G. Ciarlet, Plates and Junctions in Elastic Multi-structures: An Asymptotic Analysis, Springer-Verlag, 1990.
  • P.G. Ciarlet, Mathematical Elasticity, Theory of Shells, Vol. III, North-Holland, Amsterdam, 2000.
  • P.G. Ciarlet, The Finite Element Method for Elliptic Problems, SIAM Society for Industrial and Applied Mathematics, Philadelphia, 2002.
  • P.G. Ciarlet, Linear and Nonlinear Functional Analysis with Applications, Society for Industrial and Applied Mathematics, Philadelphia, 2013.
  • L.C. Evans, Partial Differential Equations, American Mathematical Society, Providence, 2010.
  • F. Hecht, New development in FreeFem ++, Numer. Math. 20(3-4) (2012), pp. 251–265.
  • J. Hu and Z.C. Shi, Two lower order nonconforming rectangular elements for the Reissner–Mindlin plate, Math. Comput. 76(260) (2007), pp. 1771–1787.
  • J. Hu and Z.C. Shi, Two lower order nonconforming quadrilateral elements for the Reissner–Mindlin plate, Sci. China Math. 51(11) (2008), pp. 2097–2114.
  • J.G. Huang, X.H. Huang, and W.M. Han, A new C0 discontinuous Galerkin method for Kirchhoff plates, Comput. Methods. Appl. Mech. Eng. 199(23) (2010), pp. 1446–1454.
  • A. Kristály, Fundamental tones of clamped plates in nonpositively curved spaces, Adv. Math. (N. Y) 367 (2020), pp. 107–113.
  • P. Lascaux and P. Lesaint, Some nonconforming finite elements for the plate bending problem, Rev. Dranç. D'automat. Inform. Recher. Opération. Anal. Num. 9(R1) (1975), pp. 9–53.
  • Y.L. Liao and P.B. Ming, A family of nonconforming rectangular elements for strain gradient elasticity, Adv. Appl. Math. Mech. 11(6) (2019), pp. 1263–1286.
  • N.J. Ma, R.H. Wang, Q. Han, P.J. Li, and L. Piao, Local vibration of orthotropic steel bridge decks, China J Highw. Transp. 25(6) (2012), pp. 88–93.
  • P.B. Ming and Z.C. Shi, Two nonconforming quadrilateral elements for the Reissner-Mindlin plate, Math. Models Methods Appl. Sci. 15(10) (2005), pp. 1503–1517.
  • X.Q. Shen, Q. Yang, L. Bai, and K.T. Li, Full discretization scheme for the dynamics of elliptic membrane shell model, Commun. Comput. Phys. 29(1) (2021), pp. 186–210.
  • X.Q. Shen, Y.J. Xue, Q. Yang, and S.F. Zhu, Finite element method coupling penalty method for flexural shell, Adv. Appl. Math. Mech. 14(2) (2022), pp. 365–385.
  • G. Strang and G.J. William, An Analysis of the Finite Element Method, Prentice-Hall, New York, 1973.
  • P. Trouvé, Sur la convergence des méthodes d'éléments finis nonconformes pour des problèmes linéaires de coques minces, Numer. Math. 57(1) (1990), pp. 481–524.
  • E. Ventsel and T. Krauthammer, Thin plates and shells: theory, analysis, and applications, Appl. Mech. Rev. 55(4) (2001), pp. 1813–1831.
  • R.F. Wu, X.Q. Shen, and J.K. Zhao, Conforming finite element methods for two-dimensional linearly elastic shallow shell and clamped plate models, Appl. Math. Comput. 430 (2022), pp. 127259.
  • R.F. Wu, X.Q. Shen, D.Y. Shi, and J.P. Yu, Nonconforming finite element method for two-dimensional linear elastic shallow shell model, Adv. Appl. Math. Mech. (2022). accepted.
  • X.J. Xu, L.K. Li, and W.B. Chen, A multigrid method for the mortar-type Morley element approximation of a plate bending problem, SIAM. J. Numer. Anal. 39(5) (2002), pp. 1712–1731.
  • O.P. Yadav and R. Jiwari, Finite element analysis and approximation of Burgers'-Fisher equation, Numer. Methods. Partial. Differ. Equ. 33(5) (2017), pp. 1652–1677.
  • O.P. Yadav and R. Jiwari, A finite element approach for analysis and computational modelling of coupled reaction diffusion models, Numer. Methods. Partial. Differ. Equ. 35(2) (2019), pp. 830–850.
  • O.P. Yadav and R. Jiwari, A finite element approach to capture turing patterns of autocatalytic Brusselator model, J. Math. Chem. 57(3) (2019), pp. 769–789.
  • O.P. Yadav and R. Jiwari, Some soliton-type analytical solutions and numerical simulation of nonlinear Schrödinger equation, Nonlinear. Dyn. 95(4) (2019), pp. 2825–2836.
  • G.Z. Yu, X.P. Xie, and X. Zhang, Parameter extension for combined hybrid finite element methods and application to plate bending problems, Appl. Math. Comput. 216(11) (2010), pp. 3265–3274.
  • Z.M. Zhang, Locking and robustness in the finite element method for circular arch problem, Numer. Math. 69(4) (1995), pp. 509–522.
  • Y. Zhang, Large deflection of clamped circular plate and accuracy of its approximate analytical solutions, Sci. China: Phys. Mech. Astron. 59(1) (2016), pp. 1–18.

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