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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)
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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 

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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