Keywords: post-processing, recovery techniques, superconvergence, elasticity, stability, error estimate, adaptivity.

The classical displacement formulation of the finite element method is based on a variational form of a boundary value problem in the primal variable. After solving the primal problem, the stresses are normally obtained by the appropriate constitutive equation, leading to a discontinuous and a lower order approximation. Since these quantities are of main interest in engineering problems, to improve the numerical solution of the gradient is of great importance.

The proposed recovery technique was initially formulated for Poisson's problems computed by quadrilateral element [1] and adapted for triangular element in [2]. The basic idea is to find a better approximation for the gradients by solving local variational problems. The simplicity of the computational code and the accuracy of the results stimulate further researches to more general problems. An adapted technique to improve the quality of the numerical solution in elasticity problems computed by quadrilateral elements is proposed in [3].

In this work, we extend this technique to triangular elements for plane elasticity analysis. The domain is decomposed into patch or macroelements defined as a union of neighboring elements with common edges. The macroelement post-processing technique for elasticity is based on the least squares residuals of the balance equation, kinematic equation and irrotationality condition, appropriately weighted.

The kinematic equation is evaluated by appropriate points, detected by Babuska [4,5], taking into account the superconvergence of the gradients of the displacements in the tangent direction to the boundary of the triangular element. After obtaining the post-processed gradients, a better approximation for the stresses can be calculated by the constitutive relation.

The stability and convergence of this method depend on the choice of the finite element space, on the number of elements in the patch or macroelement and on the number of superconvergence points. For a given macroelement configuration, stability is proved by solving numerically a local eigenvalue problem associated with the post-processing variational formulation. We present stability analyses of a large class of macroelements composed by linear, quadratic and cubic triangular elements. A priori error estimates with optimal rates of convergence for the gradient of the displacements and stresses are then obtained and confirmed numerically.

We also extend to elasticity problems the local error estimator initially proposed for Poisson's problems in [2]. A natural norm for the error estimator associated with the macroelement technique should include the least-square residuals of the kinematic relation, balance equation and irrotationality condition, appropriately weighted. Adaptive analysis adopting the proposed error estimator as an error indicator is presented. Numerical experiments indicate the the efficiency of the proposed error estimator associated with the macroelement post-processing technique for both linear and quadratic triangular elements.

1

A. F. D. Loula, F. A. Rochinha and M. A. Murad, "Higher-order Gradient Post-Processing for Second-Order Elliptic Problems", Comp. Methods Appl. Mech. Eng., 128, 361-381, 1995. doi:10.1016/0045-7825(95)00885-3

2

R. C. C. Silva and A. F. D. Loula, "Local Residual Error Estimator and Adaptive Finite Element Analysis of Poisson's Problems", Finite Element: Techniques and Developments, Civil-Comp Press, 87-93, Scotland, 2000. doi:10.4203/ccp.65.3.3

3

A. F. D. Loula and D. S. Pinto Jr, "Higher Order Gradient Recovery in 2-D Elasticity", Proceeding of the Fourth World Congress on Computational Mechanics (IV WCCM), Buenos Aires, 1998.

4

I. Babuska, T. Strouboulis, C. S. Upadhyay and S. K. Gangaraj, "Study of Superconvergence by Computer-Based Approach, Superconvergence of the Gradients of the Displacement, the Strain and Stress in Finite Element Solutions for Plane Elasticity", Technical Note BN-1166, Institute for Physical Science and Technology, University of Maryland, College Park, 1993.

5

I. Babuska, T. Strouboulis, C. S. Upadhyay and S. K. Gangaraj, "Computer-Based Proof of the Existence of Superconvergence Points in the Finite Element Method; Superconvergence of the Derivatives in Finite Element Solutions of Laplace's, Poisson's and Elasticity Equations", Num. Met. for Partial Differential Equations, 12, 347-392, 1996. doi:10.1002/num.1690120303

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