Element Free Galerkin Post-processing Technique Based Error Estimator for Elasticity Problems

Mohd. Ahmed, Devender Singh, M. Noor Desmukh


The study present a Mesh Free based post-processing technique for asymptotically (upper) bounded error estimator for Finite Element Analyses of elastic problems. The proposed technique uses Galerkin Element Free procedure for recovery of the displacement derivatives over a patch of nodes in radial domains. The radial nodes patches are used to construct the trial shape functions utilizing the moving least-squares (MLS) techniques. The proposed technique has been tested on three benchmark elastic problems discretized using 4-node quadrilateral elements. The recovered nodal stresses are utilized to calculate the error in finite element solution in energy norm. The study also demonstrates adaptive analysis application of proposed error estimator. The performance of proposed error estimator based on mesh independent node patches has been compared with that of mesh dependent node patches based Zienkiewicz-Zhu (ZZ) error estimator on structured and unstructured mesh. The improved results of the proposed error estimator in terms of convergence rate and effectivity are obtained. It is shown that present study incorporates the superiority of the Mesh Free Galerkin method into finite element analysis environment.


Error Estimation; Effectivity; Error Norm; Post-Processing Technique; Mesh Free Approach; Convergence; Moving Least Square Techniques.


Gerasimov, T., Ru¨ter, M. and Stein E. “An explicit residual-type error estimator for Q 1-quadrilateral extended finite element method in two-dimensional linear elastic fracture mechanics.” International Journal for Numerical Methods in Engineering, 90 (April 2012):1118–1155. doi:10.1002/nme.3363.

Ladev`eze, P. and Leguillon D. “Error estimate procedure in the finite element method and applications.” SIAM Journal on Numerical Analysis. 20(3), 1983: 485–509. doi:10.1137/0720033.

Zienkiewicz O. C. and Zhu J. Z. “Simple Error Estimator and Adaptive Procedure for Practical Engineering Analysis.” Int J. Numer. Meth. Eng., 24 (1987): 337-357. doi:10.1002/nme.1620240206.

Gratsch, T and Bathe, K. “A Posteriori error estimation techniques in practical finite element analysis,” Computers and Structures 83 (2005): 235–265. doi:1016/j.compstruc.2004.08.011.

Nadal, E… D´ıez, P., R´odenas, J.J., Tur, M. and Fuenmayor, F.J. “A recovery-explicit error estimator in energy norm for linear elasticity,” Computer Methods in Applied Mechanics and Engineering, 287 (2015): 172-190. doi:10.1016/j.cma.2015.01.013.

Hannukainen, A., Korotov, S. and R¨uter, M. “A posteriori error estimates for some problems in linear elasticity.” Helsinki University of Technology, Institute of Mathematics; Research Reports A522 (2007): 1-14. Available online: https://math.aalto.fi/reports/a522.ps.

Ramsay, A.C.A. and Maunder, E.A.W. “Effective error estimation from continuous, boundary admissible estimated stress fields.” Computers & Structures 61(2 1996): 331-343. doi:10.1016/0045-7949(96)00034-X.

Riedlbeck, Rita, Daniele A. Di Pietro, and Alexandre Ern. “Equilibrated Stress Reconstructions for Linear Elasticity Problems with Application to a Posteriori Error Analysis.” Finite Volumes for Complex Applications VIII - Methods and Theoretical Aspects (2017): 293–301. doi:10.1007/978-3-319-57397-7_22.

Yang,H. H. and Deeks,A. J. “A node-based error estimator for the element free Galerkin (EFG) Method.” International Journal of Computational Methods, 8(1 2014): 91-118. doi: 10.1142/S021987621350059X.

Zhang,G. Y., Liu, G. R. and Li, Y. “An efficient adaptive analysis procedure for certified solutions with exact bounds of strain energy for elasticity problems.” J. Finite Elem. Anal. Design, 44(14 2008): 831-841. doi:10.1016/j.finel.2008.06.010.

Cai, Difeng, and Zhiqiang Cai. “A Hybrid a Posteriori Error Estimator for Conforming Finite Element Approximations.” Computer Methods in Applied Mechanics and Engineering 339 (September 2018): 320–340. doi:10.1016/j.cma.2018.04.050.

S. Kumar, L. Fourment , S. Guerdoux, Parallel, second-order and consistent remeshing transfer operators for evolving meshes with superconvergence property on surface and volume, Finite Elements in Analysis and Design 93 (2015) 70–84. doi:10.1016/j.finel.2014.09.002

Zienkiewicz O. C. and Zhu J. Z. “The Super-convergent Patch Recovery and a posteriori Error Estimates, Part I, The Error Recovery Technique.” Int. J. Num. Meth. Engg., 33 (1992): 1331-1364. doi:10.1002/nme.1620330703.

Ahmed, M., Singh, D. and Islam, S. “Effect of Contact Conditions on Adaptive Finite Element Simulation of Sheet Forming Operations.” European Journal of Computational Mechanics, 24, (1 1015): 1-15. doi:10.1080/17797179.2015.1012632.

Parret-Fréaud, A., V. Rey, P. Gosselet, and C. Rey. “Improved Recovery of Admissible Stress in Domain Decomposition Methods - Application to Heterogeneous Structures and New Error Bounds for FETI-DP.” International Journal for Numerical Methods in Engineering 111, no. 1 (January 18, 2017): 69–87. doi:10.1002/nme.5462.

González-Estrada, O.A.; Nadal, E.; Ródenas, J.J.; Kerfriden, P.; Bordas, S.P.A.; Fuenmayor, F.J. “Mesh adaptivity driven by goal-oriented locally equilibrated super-convergent patch recovery.” Computational Mechanics, 53(2014): 957-976. doi:10.1007/s00466-013-0942-8.

Onate E. Perazzo, F. and Miquel, J. “A finite point method for elasticity problems.” Computers and Structures, 79(2001): 2151-2163. doi:10.1016/S0045-7949(01)00067-0.

Ahmed, Mohd, and Devender Singh. "An adaptive parametric study on mesh refinement during adaptive finite element simulation of sheet forming operations." Turkish Journal of Engineering and Environmental Sciences 32, no. 3 (2008): 163-175.

Rodenas, J. J., Estrada, G., Andres, O., Fuenmayor, F. J. and Chinesta, F. “Enhanced error estimator based on a nearly equilibrated moving least squares recovery technique for FEM and XFEM.” Computational Mechanics, 52 (2 2013): 321-344. doi:10.1007/s00466-012-0814-7.

Full Text: PDF

DOI: 10.28991/cej-03091211


  • There are currently no refbacks.

Copyright (c) 2018 Mohd. Ahmed, Devender Singh, M. Noor Desmukh

Creative Commons License
This work is licensed under a Creative Commons Attribution 4.0 International License.