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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:  10 November 2015

Zhengqin Yu  and
Xiaoping Xie
Zhengqin Yu
Affiliation:
School of Mathematics, Sichuan University, Chengdu 610064, China
Xiaoping Xie*
Affiliation:
School of Mathematics, Sichuan University, Chengdu 610064, China
*
*Corresponding author. Email addresses: zhengqinyulm@sina.com (Z. Yu), xpxie@scu.edu.cn (X. Xie)
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Abstract

This paper proposes and analyzes semi-discrete and fully discrete hybrid stress finite element methods for elastodynamic problems. A hybrid stress quadrilateral finite element approximation is used in the space directions. A second-order center difference is adopted in the time direction for the fully discrete scheme. Error estimates of the two schemes, as well as a stability result for the fully discrete scheme, are derived. Numerical experiments are done to verify the theoretical results.

Keywords

elastodynamic problemhybrid stress finite elementsemi-discretefully discrete

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
Numerical Mathematics: Theory, Methods and Applications , Volume 8 , Issue 4 , November 2015 , pp. 582 - 604
Copyright
Copyright © Global-Science Press 2015 

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References

[1]Becache, E., Joly, P., Tsogka, C., An analysis of new finite elements for the approximation of wave propagation problems, SIAM J. Numer. Anal. 2000, 37(4): 1503–1084.CrossRefGoogle Scholar
[2]Becache, E., Joly, P., Tsogka, C., A new family of mixed finite elements for the linear elastodynamic problem, SIAM J. Numer. Anal. 2002, 39(6): 2109–2132.CrossRefGoogle Scholar
[3]Boulaajine, L., Farhloul, M., 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. 2009, 231(1): 447–472CrossRefGoogle Scholar
[4]Boulaajine, L., Farhloul, M., 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. 2011, 235(5): 1288–1310CrossRefGoogle Scholar
[5]Brezzi, F., Fortin, M., Mixed and finite element method, Springer-Verlag, New York, 1991CrossRefGoogle Scholar
[6]Cheng, L.F., Xie, X.P., The space-time noncomforming finite element analysis for the vibration model of plane elasticity, J. Sichuan University(Natural Science Edition, in Chinese). 2012, 49(2): 258–266Google Scholar
[7]Clough, R.W., Penzien, T., Dynamics of structures, second ed., McGraw-Hill, Inc., New York, 1993Google Scholar
[8]Douglas, J. Jr. and Gupta, C.P., Superconvergence for a mixed finite element method fot elastic wave propagation in a plane domain, Numer. Math. 1986, 49(2-3): 189–202CrossRefGoogle Scholar
[9]Lai, J.J., Huang, J.G., Chen, C.M., Vibration analysis of plane elasticity problems by the C°— continuous time stepping finite element method, Appl. Numer. Math. 2009, 59(5): 905–919CrossRefGoogle Scholar
[10]Pitkäranta, J., Stenberg, R., Error bounds for the approximation of the Stokes problem using bilinear/constant elements on irregular quadrilateral meshes, in The Mathematics of Finite Elements and Applications V, Whiteman, J. R., ed., Academic Press, London, 1985: 325–334.CrossRefGoogle Scholar
[11]Pian, T.H.H., Derivation of element stiffness matrices by assumed stress distributions, ALAA J. 1964, 2(5): 1333–1336Google Scholar
[12]Pian, T.H.H., Sumihara, K., Rational approach for assumed stress finite elements, Int. J. Numer. Methods Engrg. 1984, 20(9): 1685–1695CrossRefGoogle Scholar
[13]Pian, T.H.H., Tong, P., Relations between incompatible displacement model and hybrid stress model, Int. J. Numer. Methods Engrg. 1986, 22(1): 173–181CrossRefGoogle Scholar
[14]Pian, T.H.H., Wu, C.C., A rational approach for choosing stress terms for hybrid finite element formulations, Int. J. Numer. Methods Engrg. 1988, 26(10): 2331–2343CrossRefGoogle Scholar
[15]Thomee, V., Galerkin finite element methods for parabolic problems, Springer, New York, 1997CrossRefGoogle Scholar
[16]Xie, X.P., An accurate hybrid macro-element with linear displacements, Commun. Numer. Methods Engrg. 2005, 21(1): 1–12CrossRefGoogle Scholar
[17]Xie, X.P., Zhou, T.X., Optimization of stress modes by energy compatibility for 4-node hybrid quadrilaterals, Int. J. Numer. Methods Engrg. 2004, 59(2): 293–313CrossRefGoogle Scholar
[18]Xie, X.P., Zhou, T.X., Accurate 4-node quadrilateral elements with a new version of energy-compatible stress mode, Commun. Numer. Methods Engrg. 2008, 24(2): 125–139CrossRefGoogle Scholar
[19]Yu, G.Z., Xie, X.P., Carstensen, C., Uniform convergence and a posteriori error estimation for assumed stress hybrid finite elment methods, Comput. Methods Appl. Mech. Engrg. 2011, 200(29-32): 2421–2433CrossRefGoogle Scholar
[20]Zhang, Z.M., Analysis of some quadrilateral nonconforming elements for incompressible elasiticity, SIAM J. Numer. Anal. 1997, 34(2): 640–663CrossRefGoogle Scholar
[21]Zienkiewicz, O.C., Taylor, R.L., The finite element method, vol.2, McGraw-Hill Book Company, London, 1989Google Scholar
[22]Zhou, T.X., Nie, Y.F., Combined hybrid approach to finite element schemes of high performance, Int. J. Numer. Methods Engrg. 2001, 51(2): 181–202CrossRefGoogle Scholar