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Robust Semi-Discrete and Fully Discrete Hybrid Stress Finite Element Methods for Elastodynamic Problems

  • Xiaojing Xu (a1) and Xiaoping Xie (a2)
Abstract
Abstract

This paper analyzes semi-discrete and fully discrete hybrid stress quadrilateral finite element methods for 2-dimensional linear elastodynamic problems. The methods use a 4 node hybrid stress quadrilateral element in the space discretization. In the fully discrete scheme, an implicit second-order scheme is adopted in the time discretization. We derive optimal a priori error estimates for the two schemes and an unconditional stability result for the fully discrete scheme. Numerical experiments confirm the theoretical results.

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*Corresponding author. Email: xuxiaojing0603@126.com (X. J. Xu), xpxie@scu.edu.cn (X. P. Xie)
References
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[1] Makridakis C. G., On mixed finite element methods for linear elastodynamics, Numerische Mathematik, 61(1) (1992), pp. 235260.
[2] Bécache E., Joly P. and Tsogka C., A new family of mixed finite elements for the linear elastodynamic problem, SIAM J. Numer. Anal., 39(6) (2002), pp. 21092132.
[3] Boulaajine L., Farhloul M. and Paquet L., A priori error estimation for the dual mixed finite element method of the elastodynamic problem in a polygonal domain I, J. Comput. Appl. Math., 231(1) (2009), pp. 447472.
[4] Boulaajine L., Farhloul M. and Paquet L., A priori error estimation for the dual mixed finite element method of the elastodynamic problem in a polygonal domain II, J. Comput. Appl. Math., 235(5) (2011), pp. 12881310.
[5] Lai J. J., Huang J. G. and Chen C. M., Vibration analysis of plane elasticity problems by the C0-continuous time stepping finite element method, Appl. Numer. Math., 59(5) (2009), pp. 905919.
[6] Hughes T. J. R. and Hulbert G. M., Space-time finite element methods for elastodynamics: formulations and error estimates, Comput. Methods Appl. Mech. Eng., 66(3) (1988), pp. 339363.
[7] Idesman A. V., Solution of linear elastodynamics problems with space-time finite elements on structured and unstructured meshes, Comput. Methods Appl. Mech. Eng., 196(9) (2007), pp. 17871815.
[8] Cheng L. F. and Xie X. P., The space-time noncomforming finite element analysis for the vibration model of plane elasticity, Journal of Sichuan University (Natural Science Edition, in Chinese), 49(2) (2012), pp. 258266.
[9] Pian T. H. H., Derivation of element stiffness matrices by assumed stress distributions, AIAA J., 2(5) (1964), pp. 13331336.
[10] Pian T. H. H. and Sumihara K., Rational approach for assumed stress finite elements, Int. J. Numer. Methods Eng., 20(9) (1984), pp. 16851695.
[11] Xie X. P. and Zhou T. X., Optimization of stress modes by energy compatibility for 4-node hybrid quadrilaterals, Int. J. Numer. Methods Eng., 59(2) (2004), pp. 293313.
[12] Xie X. P. and Zhou T. X., Accurate 4-node quadrilateral elements with a new version of energy-compatible stress mode, Commun. Numer. Methods Eng., 24(2) (2008), pp. 125139.
[13] Yu G. Z., Xie X. P. and Carstensen C., Uniform convergence and a posteriori error estimation for assumed stress hybrid finite elment methods, Comput. Methods Appl. Mech. Eng., 200(29) (2011), pp. 24212433.
[14] Zhou T. X. and Nie Y. F., Combined hybrid approach to finite element schemes of high performance, Int. J. Numer. Methods Eng., 51(2) (2001), pp. 181202.
[15] Yu Z. Q. and Xie X. P., Semi-discrete and fully discrete hybrid stress finite element methods for elastodynamic problems, Numerical Mathematics: Theory, Methods and Applications, 8(4) (2015), pp. 582604.
[16] Thomee V., Galerkin Finite element Methods for Parabolic Problems, Springer, New York, 1997.
[17] Zhang Z. M., Analysis of some quadrilateral nonconforming elements for incompressible elasticity, SIAM J. Numer. Anal., 34(2) (1997), pp. 640663.
[18] Brezzi F. and Fortin M., Mixed and Finite Element Method, Springer-Verlag, New York, 1991.
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Advances in Applied Mathematics and Mechanics
  • ISSN: 2070-0733
  • EISSN: 2075-1354
  • URL: /core/journals/advances-in-applied-mathematics-and-mechanics
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