Upper bounds of the error in local quantities using equilibrated and compatible finite element solutions for linear elastic problems

Computer Methods in Applied Mechanics and Engineering

Volumn 195, Issue 4-6, 2006, Pages 279-296

  de Almeida, J P Moitinho ^a   Pereira, O J B Almeida ^a  

^a INSTITUTO SUPERIOR TÉCNICO   (Portugal)

Author keywords

Adaptive refinement; Dual analysis; Error estimation; Local quantities; Verification

Indexed keywords

ADAPTIVE ALGORITHMS; APPROXIMATION THEORY; BOUNDARY VALUE PROBLEMS; ELASTICITY; ERROR ANALYSIS; ESTIMATION; FINITE ELEMENT METHOD; LINEAR EQUATIONS; MATHEMATICAL MODELS; OPTIMIZATION; PROBLEM SOLVING;

ADAPTIVE REFINEMENT; DUAL ANALYSIS; ERROR BOUND; ERROR ESTIMATION; FINITE ELEMENT ERROR INDICATORS; GREENBERG APPROACH; LINEAR ELASTICITY; LOCAL DISPLACEMENT; LOCAL QUANTITIES; WASHIZU APPROACH;

COMPUTATIONAL METHODS;

FINITE ELEMENT METHOD;

EID: 27744474569     PISSN: 00457825     EISSN: None     Source Type: Journal    
DOI: 10.1016/j.cma.2004.09.012     Document Type: Article

References (34)

1. H. Greenberg The determination of upper and lower bounds for the solution of the Dirichlet problem J. Math. Phys. 27 1948 161-182
2. K. Washizu Bounds for solutions of boundary value problems in elasticity J. Math. Phys. 32 1953 117-128
3. B. Fraeijs de Veubeke Displacement and equilibrium models in the finite element method in: O. Zienkiewicz G. Holister (Eds.) Stress Analysis 1965 Wiley
4. E. Trefftz, Ein Gegenstück zum Ritzschen Verfahren, in: Proceedings of the 2nd International Congress of Applied Mechanics, 1926, pp. 131-137.
5. W. Prager J. Synge Approximations in elasticity based on the concept of function space Quart. Appl. Math. 5 3 1947 241-269
6. J. Debongnie, A general theory of dual error bounds by finite elements, Tech. Rep. LMF/D5, University of Liège, 1983.
7. P. Ladeveze D. Leguillon Error estimate procedure in the finite element method and applications SIAM J. Numer. Anal. 20 3 1983 483-509
8. O. Pereira J. Almeida E. Maunder Adaptive methods for hybrid equilibrium finite element models Comput. Methods Appl. Mech. Engrg. 176 1999 19-39
9. M. Turner R. Clough H. Martin L. Topp Stiffness and deflection analysis of complex structures J. Aeronaut. Sci. 23 9 1956 805-854
10. D. Yang W. Kelly J. Isles A posteriori pointwise upper bound estimates in the finite element error Int. J. Numer. Methods Engrg. 36 1993 1279-1298
11. R. Albanese R. Fresa Upper and lower bounds for local electromagnetic quantities Int. J. Numer. Methods Engrg. 42 1998 499-515
12. I. Babuska A. Miller The post-processing approach in the finite element method (parts 1, 2 and 3) Int. J. Numer. Methods Engrg. 20 1984 1085-1129
13. I. Babuska T. Strouboulis C. Upadhyay S. Gangaraj A posteriori estimation and adaptive control of the pollution error in the h-version of the finite element method Int. J. Numer. Methods Engrg. 38 1995 4207-4235
14. I. Babuska T. Strouboulis S. Gangaraj Guaranteed computable bounds for the exact error in the finite element solution. Part I: One-dimensional model problem Comput. Methods Appl. Mech. Engrg. 176 1-4 1999 51-79
15. T. Strouboulis I. Babuska S. Gangaraj K. Copps D. Datta A posteriori estimation of the error in the error estimate Comput. Methods Appl. Mech. Engrg. 176 1-4 1999 387-418
16. T. Strouboulis I. Babuska S. Gangaraj Guaranteed computable bounds for the exact error in the finite element solution - part II: Bounds for the energy norm of the error in two dimensions Int. J. Numer. Methods Engrg. 47 1-3 2000 427-475
17. R. Rannacher F. Suttmeier A feed-back approach to error control in finite element methods: Application to linear elasticity Comput. Mech. 19 1997 434-446
18. J. Peraire A. Patera Bounds for linear-functional outputs of coercive partial differential equations: Local indicators and adaptive refinement in: P. Ladevèze J. Oden (Eds.) Advances in Adaptive Computational Methods in Mechanics 1998 Elsevier 199-216
19. P. Ladeveze P. Rougeot P. Blanchard J. Moreau Local error estimators for finite element linear analysis Comput. Methods Appl. Mech. Engrg. 176 1-4 1999 231-246
20. S. Prudhomme J.T. Oden T. Westermann J. Bass M. Botkin Practical methods for a posteriori error estimation in engineering applications Int. J. Numer. Methods Engrg. 56 8 2003 1193-1224
21. O. Pereira J. Almeida Majorantes do erro em grandezas locais obtidas a partir de soluçóes duais de elementos finitos in: J. Goicolea C.M. Soares M. Pastor G. Bugeda (Eds.) Métodos Numéricos En Ingeniería V 2002 SEMNI
22. O. Pereira J. Almeida Adaptive refinement for a local error bound based on duality Comput. Assisted Mech. Engrg. Sci. 10 4 2003 565-574
23. J. Almeida O. Pereira Equilibrium solutions for error estimation and adaptivity. Why, how and at what cost? in: N. Wiberg P. Diez (Eds.) Adaptive Modeling and Simulation 2003 CIMNE
24. R. Jones A generalization of the direct-stiffness method of structural analysis AIAA J. 2 5 1964 821-826
25. T. Pian P. Tong Basis of finite element methods for solid continua Int. J. Numer. Methods Engrg. 1 1969 3-28
26. J. Almeida J. Freitas Continuity conditions for finite element analysis of solids Int. J. Numer. Methods Engrg. 33 1992 845-853
27. J. Almeida J. Freitas Alternative approach to the formulation of hybrid equilibrium finite elements Comput. Struct. 40 4 1991 1043-1047
28. K. Washizu Variational Methods in Elasticity and Plasticity 1975 Pergamon Press Oxford
29. I. Babuska W. Rheinboldt A posteriori error estimates for the finite element method Int. J. Numer. Methods Engrg. 12 1978 1597-1615
30. D. Kelly J. Gago O. Zienkiewicz A posteriori error analysis and adaptive processes in the finite element method: Part I - error analysis Int. J. Numer. Methods Engrg. 19 1983 1593-1619
31. O. Zienkiewicz J. Zhu A simple error estimator and adaptive procedure for practical engineering analysis Int. J. Numer. Methods Engrg. 24 1987 337-357
32. O. Zienkiewicz J. Zhu The superconvergent patch recovery (SPR) and adaptive finite element refinement Comput. Methods Appl. Mech. Engrg. 101 1992 207-224
33. O. Pereira G. Bugeda Mesh optimality criteria and remeshing strategies for singular point problems in: M. Papadrakakis (Ed.) Innovative Computational Methods for Structural Mechanics 1998 Saxe-Coburg Publications Edinburgh
34. B. Jakobsen The Sleipner accident and its causes in: J. Robinson (Ed.) FEM Today and the Future 1993 Robinson and Associates 102-108