Hostname: page-component-78c5997874-v9fdk Total loading time: 0 Render date: 2024-11-10T14:56:55.089Z Has data issue: false hasContentIssue false
  • English
  • Français

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)
Get access

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 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

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