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

Part of: Partial differential equations, boundary value problems

Published online by Cambridge University Press:  09 January 2017

Xiaojing Xu  and
Xiaoping Xie
Xiaojing Xu
Affiliation:
School of Science, Southwest University of Science and Technology, Mianyang, Sichuan 621010, China
Xiaoping Xie*
Affiliation:
School of Mathematics, Sichuan University, Chengdu, Sichaun 610064, China
*
*Corresponding author. Email:[email protected] (X. J. Xu), [email protected] (X. P. Xie)
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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.

Keywords

Elastodynamic problemhybrid stress finite elementsemi-discretefully discreteerror estimate

MSC classification

Secondary: 65N12: Stability and convergence of numerical methods 65N15: Error bounds 65N30: Finite elements, Rayleigh-Ritz and Galerkin methods, finite methods
Type
Research Article
Information
Advances in Applied Mathematics and Mechanics , Volume 9 , Issue 2 , April 2017 , pp. 324 - 348
Copyright
Copyright © Global-Science Press 2017 

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References

[1] Makridakis, C. G., On mixed finite element methods for linear elastodynamics, Numerische Mathematik, 61(1) (1992), pp. 235–260.Google Scholar
[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. 2109–2132.Google Scholar
[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. 447–472.Google Scholar
[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. 1288–1310.Google Scholar
[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. 905–919.Google Scholar
[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. 339–363.Google Scholar
[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. 1787–1815.Google Scholar
[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. 258–266.Google Scholar
[9] Pian, T. H. H., Derivation of element stiffness matrices by assumed stress distributions, AIAA J., 2(5) (1964), pp. 1333–1336.Google Scholar
[10] Pian, T. H. H. and Sumihara, K., Rational approach for assumed stress finite elements, Int. J. Numer. Methods Eng., 20(9) (1984), pp. 1685–1695.Google Scholar
[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. 293–313.Google Scholar
[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. 125–139.Google Scholar
[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. 2421–2433.Google Scholar
[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. 181–202.Google Scholar
[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. 582–604.Google Scholar
[16] Thomee, V., Galerkin Finite element Methods for Parabolic Problems, Springer, New York, 1997.Google Scholar
[17] Zhang, Z. M., Analysis of some quadrilateral nonconforming elements for incompressible elasticity, SIAM J. Numer. Anal., 34(2) (1997), pp. 640–663.CrossRefGoogle Scholar
[18] Brezzi, F. and Fortin, M., Mixed and Finite Element Method, Springer-Verlag, New York, 1991.Google Scholar