Numerical comparison of several a posteriori error estimators for 2D stress analysis
Keywords:
discretization error, a posteriori error estimators, effectivity index, uniformity indexAbstract
The reliabilities of several a posteriori error estimaJors, including those of Gaga, Zienkiewicz-Zhu, and more recently those proposed by Beckers and Zhong are compared through a set of examples in plane elasticity. The examples range from those having analytic solutions to those having progressively stronger singularities. The examples generally use either 4 (or 8) node quadrilaterals for initial comparisons. The results of these examples indicaJe that, in certain cases, some of the error estimators are unreliable and do not appear to be asymptotically exact. Further studies are suggested to investigate the general validity of the initial conclusions.
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References
[AIN 89] Ainsworth M., Zhu J.Z., Craig A W., Zienkiewicz O.C. "Analysis of the
Zienkiewicz-Zhu a-posteriori error estimator in the finite element methcxi",
Int. J. Num. Meth. Engng., Vol.28, 2161-2174, 1989
[BAB 78] Babuska 1., Rheinboldt W.C. "A-posteriori error estimates for the finite
element methcxi", Int. J. Num. Meth. Engng., Vol.12, 1597-1659, 1978
[BAB 91] Babuska I., Rcxiriguez R. "The problem of the selection of an a-posteriori
error indicator based on smoothening techniques", Technical note BN-1126,
Institute for Physical Science and Technology, University of Maryland, USA,
[BAB 92] Babuska 1., Duran R., Rcxiriguez R. "Analysis of the efficiency of an a
posteriori error estimator for linear triangular finite elements", SIAM J. Num.
Anal., Vol.29, No.4, 947-964, 1992
[BEC 73] Beckers P. "Les fonctions de tension dans Ia methode des elements finis",
Collection des Publications de Ia Faculte des Sciences Appliquees de
l'Universite de Liege, No.41, 1973
[BEC 90] Beckers P., Zhong H.G. "Error estimators using solution defaults and their
roles in the adaptation of finite element meshes", in "FEM in the design
process", edit. Robinson J., 177-186, Proceedings of Sixth World Congress on
Finite Element Methods, Banff, Alberta, Canada, 1990
[BEC 92] Beckers P., Zhong H. G. "Influence of element distortions on the reliability of
some a-posteriori error estimators", in "New advances in computational
structural mechanics", edit. Ladeveze P., Zienkiewicz O.C., Elsevier, 177-187,
[FRA 65] Fraeijs de Veubeke B.M. "Displacement and equilibrium models in the finite
element method", in "Stress analysis", edit. Zienkiewicz O.C., Holister G.S.,
Wiley, London, 1%5
[FRA 68] Fraeijs de Veubeke B.M. "A conforming finite element for plate bending", Int.
J. Sol. Struc., Vol.4, No.1, 1968
[GAG 82] Gaga J.P. "A posteriori error analysis and adaptivity for the finite element
method", Ph.D. thesis, University of Wales, Swansea, U.K, 1982
[KEL 83] Kelly D.W., Gaga J.P., Zienkiewicz O.C., Babuska I. "A posteriori error
analysis and adaptive processes in the finite element method: Part I-Error
analysis", Int. J. Num. Meth. Engng., Vol.19, 1593-1619, 1983
[KEL 84] Kelly D. W. "The self -equilibration of residuals and complementary a posteriori
error estimates in the finite element method", Int. J. Num. Meth. Engng.,
Vol.20, 1491-1506, 1984
[KIK 86] Kikuchi N. "Adaptive grid design for finite element analysis in optimization:
Part I. Review of finite element error analysis; Part II. Grid optimization;
Part III. Shape optimization", in "Computer aided optimal design, structural
and mechanical systems" (Vol.2), edit. Mota Soares C.A, 307-363, 1986
[LAD 77] Ladeveze P. "Nouvelle procedure d'estimation d'erreur relative a Ia methode
des elements finis et applications", Journees Elements Finis, Rennes, France,
[LAD 83] Ladeveze P., Leguillon D. "Error estimate procedure in the finite element
method and applications", SIAM J. Num. Anal., Vol.20, No.3, 483-509, 1983
[MAU 90] Maunder E.A W., Hill W.G. "Complementary use of displacement and
equilibrium models in analysis and design", in "FEM in the design process",
edit. Robinson J., 493-501, Proceedings of Sixth World Congress on Finite
Element Methods, Banff, Alberta, Canada, 1990
[ODE 71] Oden J.T., Brauchli H.T. "On the calculation of consistent stress distributions
in finite element calculations", Int. J. Num. Meth. Engng., Vol.3, 317-325,
[ODE 86] Oden J.T., Strouboulis T., Devloo P. "Adaptive methods for problems in solid
and fluid mechanics", in "Accuracy estimates and adaptive refinements in
finite element computations", edit. Babuska I., Zienkiewicz O.C., Gaga J.P.,
Oliveira E.R. de A, Wiley, New York, 249-280, 1986
[ODE 89] Oden J.T., Demkowicz L., Rochowicz W., Westermann T.A "Toward a
universal h-p adaptive finite element strategy, Part 2. A posteriori error
estimation", Camp. Meth. Appl. Mech. Engng., Vol.77, 113-180, 1989
[OHT 90] Ohtsubo H., Kitamura M. "Element by element a posteriori error estimation
and improvement of stress solutions for two-dimensional elastic problems".
Int. J. Num. Meth. Engng., Vol.29, 233-244, 1990
[RIC 10] Richardson L.F. "The approximate arithmetical solution by finite differences
of physical problems", Trans. Roy. Soc. (London) A210, 307-357, 1910
[ROB 92] Robinson J., Maunder E.A W., Ramsay ACA "Some studies of simple error
estimators", Parts I, II, ill, finite element news, issues 4, 5, 6, 1992
[ROB 93] Robinson J., Maunder E.A W., Ramsay AC.A "Some studies of simple error
estimators", Parts IV, V, finite element news, issues 1, 2, 1992
[ROU 89] Rougeot P. "Sur le contrOle de Ia qualite des maillages elements finis", Ph.D.
thesis, University of Paris VI, France, 1989
[SHE 89] Shephard M.S., Niu Q., Baehmann P.L. "Some results using stress projectors
for error indication and estimation", in "Adaptive methods for partial
differential equations", edit. Flaherty J.E., Paslow P J., Shephard M.S.,
Vasilakis J.D., SIAM, Philadelphia, 83-99, 1989
[STR 92] Strouboulis T., Haque K.A "Recent experiences with error estimation and
adaptivity, Part I: Review of error estimators for scalar elliptic problems",
Camp. Meth. Appl. Mech. Engng., Vol.97, 399-436, 1992
[SZA 86] Szabo B.A "Estimation and control of error based on p-convergence", in
"Accuracy estimates and adaptive refinements in finite element computations",
edit. Babusk:a I., Zienldewicz O.C., Gaga J.P., Oliveira E.R. de A, Wiley,
New York:, 61-78, 1986
[ZHO 90a] Zhong H.G., Beckers P. "Equilibrium default error estimators for the finite
element solution", Int. Rep. SA-139, LTAS, University of Liege, Belgium,
[ZHO 90b] Zhong H.G., Beckers P. "Solution approximation error estimators for the
finite element solution", Int. Rep. SA-140, LTAS, University of Liege,
Belgium, 1990
[ZHO 90c] Zhong H.G., Beckers P. "Error estimators for quadratic and bi-quadratic
elements", Proceedings of StruCoMe 90, Paris, France, 643-657, 1990
[ZHO 91a] Zhong H.G., Beckers P. "Comparison of different a posteriori error
estimators", Int. Rep. SA-156, LTAS, University of Liege, Belgium, 1991
[ZHO 91b] Zhong H.G. "Estimateurs d'erreur a posteriori et adaptation de maillages
dans Ia methode des elements finis", Ph.D. thesis, University of Liege,
Belgium, 1991
[ZIE 87] Zienldewicz O.C., Zhu J .z. "A simple error estimator and adaptive procedure
for practical engineering analysis", Int. J. Num. Meth. Engng., Vol.24, 337-
, 1987
[ZIE 92] Zienldewicz O.C., Zhu J.Z. "The superconvergent patch recovery and a
posteriori error estimates. Part 1: The recovery technique; Part 2: Error
estimates and adaptivity", Int. J. Num. Meth. Engng., Vol.33, 1331-1382,
