Hostname: page-component-586b7cd67f-t8hqh Total loading time: 0 Render date: 2024-11-28T01:06:22.428Z Has data issue: false hasContentIssue false
  • English
  • Français

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)
Get access

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 

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] 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