An improved stress recovery technique for low‐order 3D finite elements

Abstract

SummaryIn this paper, we propose a stress recovery procedure for low-order finite elements in 3D. For each finite element, the recovered stress field is obtained by satisfying equilibrium in an average sense and by projecting the directly calculated stress field onto a conveniently chosen space. Compared with existing recovery techniques, the current procedure gives more accurate stress fields, is simpler to implement, and can be applied to different types of elements without further modification. We demonstrate, through a set of examples in linear elasticity, that the recovered stresses converge at a higher rate than that of directly calculated stresses and that, in some cases, the rate of convergence is the same as that of the displacement field.

Smoothed numerical manifold method with physical patch‐based smoothing domains for linear elasticity

“…Note that numerous experience manifests that the convergences of some other SFEMs such as the ES-FEM are often higher. Besides the enhanced accuracy and convergence, which are produced by other SFEMs and some postprocessing schemes 37 as well, the unique quasi-equilibrium characteristics makes NS-FEM appealing in search of the exact solutions and appropriate for the nearly incompressible solids. 40 For more detailed fundamentals and applications of SFEM, we refer to the monograph 47 and the review article 52 as well as the references therein.…”

“…35 Moreover, the notorious linear dependence problem accompanying high-order NMM brings about challenges in solving the linear equations. Another approach commonly employed to enhance the performance of the low-order approximation is the recovery 36,37 designed to improve the accuracy of stresses, whereas it is incapable of overcoming the volumetric locking and stress oscillations in analyzing the nearly incompressible solid. Complicated techniques such as the variational multiscale method 38 have to be adopted to resolve the volumetric locking and stress oscillations.…”

Efficient and accurate stress recovery procedure and a posteriori error estimator for the stable generalized/extended finite element method

“…1 Global projections are computationally expensive and nontrivial to implement. 2 More computationally efficient methods include the superconvergent patch recovery (SPR) method of Zienkiewicz and Zhu, 3,4 the recovery by equilibrium in patched method by Boroomand and Zienkiewicz,5,6 the nodal-point force method by Payen and Bathe, 7,8 and, more recently, the stress improvement procedure method by Payen and Bathe 9 and by Sharma et al 2 Further details on these methods can be found in the work of Sharma et al 2 Stress recovery methods for the generalized/extended FE method (G/XFEM) [10][11][12][13] and for the stable generalized FEM (SGFEM) 14,15 are far less mature than for the FEM. The approximation spaces of these methods in general involve nonpolynomial functions (cf Section 3), and this must be taken into account by stress recovery techniques for these methods.…”

“…Nodal averaging and global L 2 projection of FE stress are among the earliest methods used for stress smoothing and recovery in the FEM . Global projections are computationally expensive and nontrivial to implement . More computationally efficient methods include the superconvergent patch recovery (SPR) method of Zienkiewicz and Zhu, the recovery by equilibrium in patched method by Boroomand and Zienkiewicz, the nodal‐point force method by Payen and Bathe, and, more recently, the stress improvement procedure method by Payen and Bathe and by Sharma et al Further details on these methods can be found in the work of Sharma et al…”

A New Nodal Stress Recovery Technique in Finite Element Method Using Colliding Bodies Optimization Algorithm

“…Anitescu et al [13] executed a recovery based error estimation technique in high-order splines. For the low order 3D FEM, Sharma et al [14] improved stress recovery. Wang et al [15] employed superconvergence analysis for solving Maxwell's equations and Yuan et al [16] developed a new adaptive finite element method (AFEM) based on element energy projection method.…”