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Comparison of error estimation methods and adaptivity for plane stress/strain problems

  • Ozakca, Mustafa (Department of Civil Engineering, University of Gaziantep)
  • Received : 2001.11.13
  • Accepted : 2003.02.21
  • Published : 2003.05.25

Abstract

This paper deals with adaptive finite element analysis of linearly elastic structures using different error estimators based on flux projection (or best guess stress values) and residual methods. Presentations are given on a typical h-type adaptive analysis, a mesh refinement scheme and the coupling of adaptive finite element analysis with automatic mesh generation. Details about different error estimators are provided and their performance, reliability and convergence are studied using six node quadratic triangular elements. Several examples are presented to demonstrate the reliability of different error estimators.

Keywords

References

  1. Babuska, I. and Rheinboldt, W.C. (1978), "A posteriori error estimates for the finite element method", Int. J. Num. Meth. Engng., 12, 1597-1615. https://doi.org/10.1002/nme.1620121010
  2. Babuska, I. and Yu, D. (1986), "Asymptotically exact a posteriori error estimator for biquadratic elements", Technical Note BN-1050, Institute for Physical Science and Technology, University of Maryland.
  3. Babuska, I., Strouboulis, T. and Upadhyay, C.S. (1994), "A model study of the quality of a posteriori error estimators for linear elliptic problems, Error estimation in the interior of patchwise uniform grids of triangles", Comput. Meth. Appl. Mech. Eng., 114, 307-378. https://doi.org/10.1016/0045-7825(94)90177-5
  4. Babuska, I., Strouboulis, T., Upadhyay, C.S., Gangaraj, S.K. and Copps, K. (1994), "Validation of a posteriori error estimators by numerical approach", Int. J. Num. Meth. Engng., 37, 1073-1123. https://doi.org/10.1002/nme.1620370702
  5. Baehmann, P.L., Shephard, M.S. and Flaherty, E.J. (1990), "A posteriori error estimation for triangular and tetrahedral quadratic elements using interior residuals", SCOREC Report, Scientific Computation Research Center, Rensselaer Polytechnic Institute.
  6. Barlow, J. (1976), "Optimal stress locations in the finite element method", Int. J. Num. Meth. Engng., 10, 243-251. https://doi.org/10.1002/nme.1620100202
  7. Cantin, G., Loubignac, G. and Touzot, G. (1978), "An iterative algorithm to build continuous stress and displacement solutions", Int. J. Num. Meth. Engng., 12, 1493-1506. https://doi.org/10.1002/nme.1620121004
  8. Cho, J.R and Oden, J.P. (1996), "A priori error estimations of hp - finite element approximations for hierarchical models of plate and shell like structures", Comp. Meth. Appl. Mech. Eng., 132, 135-177. https://doi.org/10.1016/0045-7825(95)00985-X
  9. Cook, R.D., Malkus, S.D. and Plesha, M.E. (1989), Concepts and Applications of Finite Element Analysis, 3th ed. John Wiley, New York.
  10. Cugnon, F. and Beckers, P. (1998), "Error estimation for h- and p-method", 8th Mechanical Engineering Chilean Congress 183-188, 27-30 October.
  11. Dutta, A. and Ramakrishnan, C.V. (1997), "Error estimation in finite element transient dynamic analysis using modal superposition", Engineering Computation, 14, 135-158. https://doi.org/10.1108/02644409710157668
  12. Hinton, E. and Campbell, J.S. (1974), "Local and global smoothing of discontinuous finite element functions using a least squares method", Int. J. Num. Meth. Engng., 8, 461-480. https://doi.org/10.1002/nme.1620080303
  13. Hinton, E., Rock, T. and Zienkiewicz, O.C. (1976), "A note on mass lumping and related processes in the finite element method", Int. J. Earth. Engng. Struct. Dyn., 4, 245-249. https://doi.org/10.1002/eqe.4290040305
  14. Hinton, E., Ozakca, M. and Rao, N.V.R. (1991), "An integrated approach to structural shape optimisation of linearly elastic structures Part 2: shape definition and adaptivity", Computing Systems in Engineering, 2, 27-56. https://doi.org/10.1016/0956-0521(91)90037-6
  15. Hinton, E., Özakça, M. and Rao, N.V.R. (1991), "Adaptive analysis of thin shells using facet elements", Int. J. Num. Meth. Engng., 32, 1283-1301. https://doi.org/10.1002/nme.1620320608
  16. Kelly, D.W., Gago, J.P. de S.R. and Zienkiewicz, O.C. (1983), "A posteriori error analysis and adaptive processes in the finite element method, Part I: error analysis", Int. J. Num. Meth. Engng., 19, 1593-1619. https://doi.org/10.1002/nme.1620191103
  17. Mathisen, K.M and Okstad, K.M. (1999), "Error estimation and adaptivity in explicit nonlinear finite element simulation of quasi-static problems", Comput. Struct., 72 , 627-644. https://doi.org/10.1016/S0045-7949(98)00328-9
  18. Moan, T. (1974), "Optimal polynomials and 'best' numerical integration formulas on a triangle", ZAMM, 54, 501-508. https://doi.org/10.1002/zamm.19740540706
  19. Onate, E., Castro, J. and Kreiner, R. (1992), "Error estimation and mesh adaptivity techniques for plate and shell problems", Proc. of the Third Int. Conf. on Quality Assurance and Standards in Finite Element Analysis, 1-17.
  20. Ozakca, M. (1993), "Analysis and optimal design of structures with adaptivity", Ph.D thesis, C/Ph/168/93, Dept. of Civil Engineering, University College of Swansea, Swansea, UK.
  21. Peraire, J., Vahdati, M., Morgan, K. and Zienkiewicz, O.C. (1987), "Adaptive remeshing for compressible flow computations", J. Comp. Phys., 72, 449-466. https://doi.org/10.1016/0021-9991(87)90093-3
  22. Robinson, D.J. and Armstrong, C.G. (1992), "An experimental comparison of energy error estimators for 8-noded isoparametric quadrilateral elements", Proc. of the Third Int. Conf. on Quality Assurance and Standards in Finite Element Analysis, NAFEM, 202-212.
  23. Shephard, M.S., Niu, Q.X. and Baehmann, P.L. (1989), "Some results using stress projectors for error indication and estimation", SCOREC Report, Scientific Computation Research Center, Rensselaer Polytechnic Institute.
  24. Sienz, J. (1994), "Integrated structural modelling, adaptive analysis and shape optimization", Ph.D. thesis, C/M/ 259/90, Dept. of Civil Eng. University College of Swansea.
  25. Stephen, D.B and Steven, G.P. (1997), "Natural frequency error estimation using a path recovery technique", J. Sound Vib., 200, 151-165. https://doi.org/10.1006/jsvi.1996.0658
  26. Strouboulis, T. and Haque, K.A. (1992), "Recent experiences with error estimation and adaptivity, Part 2: error estimation for h - adaptive approximations on grids of triangles and quadrilaterals", Comp. Meth. Appl. Mech. Engng., 100, 359-430. https://doi.org/10.1016/0045-7825(92)90090-7
  27. Zienkiewicz, O.C. and Zhu, J.Z. (1992), "The superconvergent patch recovery and a posteriori error estimates. Parts 1-2", Int. J. Num. Meth. Engng., 33, 1331-1382. https://doi.org/10.1002/nme.1620330702
  28. Zienkiewicz, O.C., Liu, Y.C. and Huang, G.C. (1988), "An error estimate and adaptive refinement method for extrusion and other forming problems", Int. J. Num. Meth. Engng., 25, 23-42. https://doi.org/10.1002/nme.1620250105
  29. Zienkiewicz, O.C. and Taylor, R.L. (2000), The Finite Element Method, 5th ed., Vols. 1-3, Butterworth Heinemann, London.

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