Uniform convergence and a posteriori error estimation for assumed stress hybrid finite element methods

• Published in: Computer methods in applied mechanics and engineering Vol. 200; no. 29; pp. 2421 - 2433
• Main Authors: Yu, Guozhu, Xie, Xiaoping, Carstensen, Carsten
• Format: Journal Article
• Language: English
• Published: Kidlington Elsevier B.V 01.07.2011; Elsevier
• Subjects: A posteriori estimator; Assumed stress hybrid method; Displacement; Errors; Exact sciences and technology; Finite element; Finite element method; Fundamental areas of phenomenology (including applications); Hellinger–Reissner principle; INT; Mathematical analysis; Mathematical models; Physics; Poisson-locking; Quadrilaterals; Solid mechanics; Static elasticity (thermoelasticity...); Stresses; Structural and continuum mechanics; Hellinger–Reissner principle; Assumed stress hybrid method; Poisson-locking; A posteriori estimator; Finite element; Quadrilateral finite element; Isoparametric finite element; Error bound; Large displacement; Experimental study; Modeling; Optimization; Finite element method; Hellinger Reissner model; A posteriori estimation; A priori estimation; Variational calculus; Hybrid finite element; Hellinger-Reissner principle
• ISSN: 0045-7825; 1879-2138
• DOI: 10.1016/j.cma.2011.03.018

Assumed stress hybrid methods are known to improve the performance of standard displacement-based finite elements and are widely used in computational mechanics. The methods are based on the Hellinger–Reissner variational principle for the displacement and stress variables. This work analyzes two existing 4-node hybrid stress quadrilateral elements due to Pian and Sumihara [T.H.H. Pian, K. Sumihara, Rational approach for assumed stress finite elements, Int. J. Numer. Methods Engrg. 20 (9) (1984) 1685–1695] and due to Xie and Zhou [X.P. Xie, T.X. Zhou, Optimization of stress modes by energy compatibility for 4-node hybrid quadrilaterals, Int. J. Numer. Methods Engrg. 59 (2004) 293–313], which behave robustly in numerical benchmark tests. For the finite elements, the isoparametric bilinear interpolation is used for the displacement approximation, while different piecewise-independent 5-parameter modes are employed for the stress approximation. We show that the two schemes are free from Poisson-locking, in the sense that the error bound in the a priori estimate is independent of the relevant Lamé constant λ. We also establish the equivalence of the methods to two assumed enhanced strain schemes. Finally, we derive reliable and efficient residual-based a posteriori error estimators for the stress in L 2-norm and the displacement in H 1-norm, and verify the theoretical results by some numerical experiments.