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

TL;DR

This paper analyzes two hybrid stress finite element methods, proving their robustness against Poisson-locking, establishing their equivalence to enhanced strain schemes, and deriving reliable a posteriori error estimators validated by numerical tests.

Contribution

It provides a rigorous convergence analysis, demonstrates the absence of Poisson-locking, and introduces effective a posteriori error estimators for specific hybrid finite element methods.

Findings

The methods are free from Poisson-locking, with error bounds independent of the Lame constant.

The schemes are equivalent to certain enhanced strain methods.

Reliable residual-based a posteriori error estimators are derived and validated.

Abstract

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 [Int. J. Numer. Meth. Engng, 1984] and due to Xie and Zhou [Int. J. Numer. Meth. Engng, 2004], 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 Lame constant $\lambda$. 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.