Hostname: page-component-586b7cd67f-rdxmf Total loading time: 0 Render date: 2024-11-28T02:10:50.706Z Has data issue: false hasContentIssue false
  • English
  • Français

Uniform Convergence Analysis of a Higher Order Hybrid Stress Quadrilateral Finite Element Method for Linear Elasticity Problems

Part of: Partial differential equations, boundary value problems

Published online by Cambridge University Press:  27 January 2016

Yanhong Bai ,
Yongke Wu  and
Xiaoping Xie
Yanhong Bai
Affiliation:
School of Mathematics, Sichuan University, Chengdu 610064, China
Yongke Wu
Affiliation:
School of Mathematical Sciences, University of Electronic Science and Technology of China, Chengdu 611731, China Institute of Structure Mechanical, China Academy of Engineering Physics, Mianyang 621900, China
Xiaoping Xie*
Affiliation:
School of Mathematics, Sichuan University, Chengdu 610064, China
*
*Corresponding author. Email: [email protected] (Y. H. Bai), [email protected] (Y. K. Wu), [email protected] (X. P. Xie)
Get access

Abstract.

This paper derives a higher order hybrid stress finite element method on quadrilateral meshes for linear plane elasticity problems. The method employs continuous piecewise bi-quadratic functions in local coordinates to approximate the displacement vector and a piecewise-independent 15-parameter mode to approximate the stress tensor. Error estimation shows that the method is free from Poisson-locking and has second-order accuracy in the energy norm. Numerical experiments confirm the theoretical results.

Keywords

Linear elasticityhybrid stress finite elementPoisson-lockingsecond-order accuracy

MSC classification

Secondary: 65N12: Stability and convergence of numerical methods 65N30: Finite elements, Rayleigh-Ritz and Galerkin methods, finite methods
Type
Research Article
Information
Advances in Applied Mathematics and Mechanics , Volume 8 , Issue 3 , June 2016 , pp. 399 - 425
Copyright
Copyright © Global-Science Press 2016 

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]Arnold, D. N., Boffi, D. and Falk, R., Approximation of quadrilateral finite elements, Math. Comput., 71 (2002), pp. 909–922.Google Scholar
[2]Boffi, D. and Gastaldi, L., On the quadrilateral Q2-P1 element for the Stokes problem, Int. J. Numer. Meth. Fluids, 39 (2002), pp. 1001–1011.CrossRefGoogle Scholar
[3]Brezzi, F. and Fortin, M., Mixed and Hybrid Finite Element Methods, Springer-Verlag, 1991.CrossRefGoogle Scholar
[4]Cook, R. D., A plane hybrid element with rotational D.O.F. and adjustable stiffness, Int. J. Numer. Meth. Eng., 24 (1987), pp. 1499–1508.Google Scholar
[5]Elman, H. C., Silvester, D. J. and Wathen, A. J., Finite Elements and Fast Iterative Solvers: with Applications in Incompressible Fluid Dynamics, Oxford University Press, 2005.Google Scholar
[6]Pian, T. H. H., Derivation of element stifness matrices by assumed stress distributions, AIAA J., 2 (1964), pp. 1333–1336.CrossRefGoogle Scholar
[7]Pian, T. H. H. and Chen, D. P., Alternative ways of for formulation of hybrid stress elements, Int. J. Numer. Meths. Eng., 18 (1982), pp. 1679–1684.Google Scholar
[8]Pian, T. H. H. and Sumihara, K., Rational approach for assumed stress finite elements, Int. J. Numer. Meth. Eng., 20 (1984), pp. 1685–1695.CrossRefGoogle Scholar
[9]Pian, T. H. H. and Tong, P., Relations between incompatible displacement model and hybrid stress model, Int. J. Numer. Meth. Eng., 22 (1986), pp. 173–181.Google Scholar
[10]Pian, T. H. H. and Wu, C. C., A rational approach for choosing stress term of hybrid finite element formulations, Int. J. Numer. Meth. Eng., 26 (1988), pp. 2331–2343.Google Scholar
[11]Shi, Z. C., A convergence condition for quadrilateral Wilson element, Numer. Math., 44 (1984), pp. 349–361.Google Scholar
[12]Spilker, R. L. and Singh, S. P., Three-dimensional hybrid-stress isoparametric quadratic displacement element, Int. J. Numer. Meth. Eng., 18 (1982), pp. 445–465.Google Scholar
[13]Sze, K. Y., Chow, C. L. and Chen, W. J., A rational formulation of isoparametric hybrid stress elements for three-dimensional stress analysis, Finite Element Anal. Design, 7 (1990), pp. 61–72.Google Scholar
[14]Sze, K. Y., Hybrid hexahedral element for solids, plates, shells and beams by selective scaling, Int. J. Numer. Methods. Eng., 36 (1993), pp. 1519–1540.CrossRefGoogle Scholar
[15]Sze, K. Y., Admissible matrix formulation-from orthobonal approach to explicit hybrid stabilization, Finite Element Anal. Design, 24 (1996), pp. 1–30.CrossRefGoogle Scholar
[16]Sze, K. Y. and Fan, H., An economical assumed stress brick element and its application, Finite Element Anal. Design, 21 (1996), pp. 179–200.Google Scholar
[17]Xie, X. P. and Zhou, T. X., Optimization of stress modes by energy copatibility for 4-node hybrid quadrilaterals, Int. J. Numer. Meth. Eng., 59 (2004), pp. 293–313.Google Scholar
[18]Xie, X. P., An accurate hybrid macro-element with linear displacements, Commum. Numer.Meth. Eng., 21 (2005), pp. 1–12.Google Scholar
[19]Xie, X. P. and Zhou, T. X., Accurate 4-node quadrilateral elements with a new version of energy- compatible stress mode, Commun. Numer. Meth. Eng., 24 (2008), pp. 125–139.CrossRefGoogle Scholar
[20]Yu, G. Z., Xie, X. P. and Carstensen, C., Uniform convergence and a posterior error estimation for assumed stress hybrid finite element methods, Comput. Methods Appl. Mech. Eng., 200 (2011), pp. 2421–2433.CrossRefGoogle Scholar
[21]Zhang, S. Q. and Xie, X.P., Accurate 8-Node hybrid hexahedral elements energy-compatible stress modes, Adv. Appl. Math. Mech., 2 (2010), pp. 333–354.Google Scholar
[22]Zhang, Z. M., Analysis of some quadrilateral nonconforming elements for incompressible elasticity, SIAM J. Numer. Anal., 34 (1997), pp. 640–663.CrossRefGoogle Scholar
[23]Zhou, T. X. and Nie, Y. F., A combined hybrid approach to finite element schemes of high performance, Int. J. Numer. Meth. Eng., 51 (2001), pp. 181–202.Google Scholar
[24]Zhou, T. X. and Xie, X. P., A unified analysis for stress/strain hybrid methods of high performance, Comput. Methods Appl. Mech. Eng., 191 (2002), pp. 4619–4640.Google Scholar