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

摘要

Assumed stress hybrid methods are known to improve the performance ofstandard displacement-based finite elements and are widely used incomputational mechanics. The methods are based on the Hellinger-Reissnervariational principle for the displacement and stress variables. This workanalyzes two existing 4-node hybrid stress quadrilateral elements due to Pianand Sumihara [Int. J. Numer. Meth. Engng, 1984] and due to Xie and Zhou [Int.J. Numer. Meth. Engng, 2004], which behave robustly in numerical benchmarktests. For the finite elements, the isoparametric bilinear interpolation isused for the displacement approximation, while different piecewise-independent5-parameter modes are employed for the stress approximation. We show that thetwo schemes are free from Poisson-locking, in the sense that the error bound inthe a priori estimate is independent of the relevant Lame constant $\lambda$.We also establish the equivalence of the methods to two assumed enhanced strainschemes. Finally, we derive reliable and efficient residual-based a posteriorierror estimators for the stress in $L^{2}$-norm and the displacement in$H^{1}$-norm, and verify the theoretical results by some numerical experiments.