A posteriori error estimation in large-strain elasticity using equilibrated local Neumann problems

Computer Methods in Applied Mechanics and Engineering

Volumn 161, Issue 1-2, 1998, Pages 77-101

  Brink, Ulrich ^a   Stein, Erwin ^a  

^a LEIBNIZ UNIVERSITY HANNOVER   (Germany)

Author keywords

[No Author keywords available]

Indexed keywords

BOUNDARY VALUE PROBLEMS; ELASTICITY; ERRORS; ESTIMATION; LINEAR EQUATIONS; PARTIAL DIFFERENTIAL EQUATIONS; PROBLEM SOLVING; STRAIN; STRESSES; TENSORS;

ERROR ESTIMATION; LARGE STRAIN ELASTICITY; NEUMANN PROBLEM;

FINITE ELEMENT METHOD;

EID: 0032479720     PISSN: 00457825     EISSN: None     Source Type: Journal    
DOI: 10.1016/S0045-7825(97)00310-1     Document Type: Article

References (38)

1. [1] M. Ainsworth and J.T. Oden, A unified approach to a posteriori error estimation using element residual methods, Numer. Math. 65 (1993) 23-50.
2. [2] M. Ainsworth and J.T. Oden, A posteriori error estimators for the Stokes and Oseen equations, SIAM J. Numer. Anal. 34 (1997) 228-245.
3. [3] M. Ainsworth and J.T. Oden, A posteriori error estimation in finite element analysis, Comput. Methods Appl. Mech. Engrg. 142 (1997) 1-88.
4. [4] M. Ainsworth, J.T. Oden and W. Wu, A posteriori error estimation for h - p approximations in elastostatics, Appl. Numer. Math. 14 (1994) 23-54.
5. [5] I. Babuška, T. Strouboulis, A. Mathur and C.S. Upadhyay, Pollution-error in the h-version of the finite-element method and the local quality of a posteriori error estimators, Finite Elem. Anal. Des. 17 (1994) 273-321.
6. [6] I. Babuška, T. Strouboulis and C.S. Upadhyay, 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. Methods Appl. Mech. Engrg. 114 (1994) 307-378.
7. [7] I. Babuška, T. Strouboulis, C.S. Upadhyay and S.K. Gangaraj, A model study of element residual estimators for linear elliptic problems: the quality of the estimators in the interior of meshes of triangles and quadrilaterals, Comput. Struct. 57 (1995) 1009-1028.
8. [8] I. Babuška, T. Strouboulis, C.S. Upadhyay, S.K. Gangaraj, K. Copps, Validation of a posteriori error estimators by numerical approach, Int. J. Numer. Meth. Engrg. 37 (1994) 1073-1123.
9. [9] R.E. Bank and A. Weiser, Some a posteriori error estimators for elliptic partial differential equations, Math. Comput. 44 (1985) 283-301.
10. [10] R.E. Bank and B.D. Welfert, A posteriori error estimates for the Stokes equations: a comparison, Comput. Methods Appl. Mech. Engrg. 82 (1990) 323-340.
11. [11] P.A.J. van den Bogert, R. de Borst, G.T. Luiten and J. Zeilmaker, Robust finite elements for 3D - analysis of rubber-like materials, Engrg. Comput. 8 (1991) 3-17.
12. [12] B. Boroomand and O.C. Zienkiewicz, Recovery by equilibrium in patches (REP), Int. J. Numer. Meth. Engrg. 40 (1997) 137-164.
13. [13] F. Brezzi, On the existence, uniqueness and approximation of saddle-point problems arising from Lagrangian multipliers, R.A.I.R.O. Anal. Numer. 8 (1974) 129-151.
14. [14] F. Brezzi and M. Fortin, Mixed and Hybrid Finite Element Methods (Springer, New York, 1991).
15. [15] U. Brink, Adaptive gemischte finite Elemente in der nichtlinearen Elastostatik und deren Kopplung mit Randelementen, Doctoral Thesis, Institut für Baumechanik und Numerische Mechanik (IBNM), Universität Hannover, 1997.
16. [16] U. Brink and E. Stein, On some mixed finite element methods for incompressible and nearly incompressible finite elasticity, Comput. Mech, 19 (1996) 105-119.
17. [17] H. Bufler and E. Stein, Zur Plattenberechnung mittels Finiter Elemente, Ing. Archiv. 39 (1970) 248-260.
18. [18] T.Y.P. Chang, A.F. Saleeb and G. Li, Large strain analysis of rubber-like materials based on a perturbed Lagrangian variational principle, Comput. Mech. 8 (1991) 221-233.
19. [19] J.S. Chen, W. Han, C.T. Wu and W. Duan, On the perturbed Lagrangian formulation for nearly incompressible and incompressible hyperelasticity, Comput. Methods Appl. Mech. Engrg. 142 (1997) 335-351.
20. [20] J.S. Chen, K. Satyamurthy and L. R. Hirschfelt, Consistent finite element procedures for nonlinear rubber elasticity with a higher order strain energy function, Comput. Struct. 50 (1994) 715-727.
21. [21] P.G. Ciarlet, Mathematical Elasticity, Vol. I: Three-Dimensional Elasticity (North-Holland, Amsterdam, 1988).
22. [22] B.Häggblad and J.A. Sundberg, Large strain solutions of rubber components, Comput. Struct. 17 (1983) 835-843.
23. [23] L.R. Herrmann, Elasticity equations for incompressible and nearly incompressible materials by a variational theorem, AIAA J. 3 (1965) 1896-1900.
24. [24] A. Huerta, P. Díez and J.J. Egozcue, An a posteriori error estimator for classical linear and nonlinear finite elements, in: J.-A. Désidéri et al. (eds.), Numerical Methods in Engineering '96, Proceedings of the 2nd ECCOMAS Conference (Wiley, Chichester, 1996) 277-283.
25. [25] P. Ladevèze and D. Leguillon, Error estimate procedure in the finite element method and applications, SIAM J. Numer. Anal. 20 (1983) 485-509.
26. [26] P. Ladevèze, P. Marin, J.P. Pelle and J.L. Gastine, Accuracy and optimal meshes in finite element computation for nearly incompressible materials, Comput. Methods Appl. Mech. Engrg. 94 (1992) 303-315.
27. [27] P. Le Tallec, Numerical methods for nonlinear three-dimensional elasticity, in: P.G. Ciarlet and J.L. Lions (eds.), Handbook of Numerical Analysis, Vol. III (Elsevier, Amsterdam, 1994) 465-622.
28. [28] J.T. Oden, W. Wu and M. Ainsworth, An a posteriori error estimate for finite element approximations of the Navier-Stokes equations, Comput. Methods Appl. Mech. Engrg. 111 (1994) 185-202.
29. [29] R.W. Ogden, Non-Linear Elastic Deformations (Ellis Horwood-John Wiley, Chicester, 1984).
30. [30] W.C. Rheinboldt and J.-L. Liu, A posteriori error estimates for parametrized nonlinear equations, in: P. Wriggers and W. Wagner (eds,), Nonlinear Computational Mechanics (Springer, Berlin, 1991) 31-46.
31. [31] J.C. Simo and R.L. Taylor, Quasi-incompressible finite elasticity in principal stretches. Continuum basis and numerical algorithms, Comput. Methods Appl. Mech. Engrg. 85 (1991) 273-310.
32. [32] J.C. Simo, R.L. Taylor and K.S. Pister, Variational and projection methods for the volume constraint in finite deformation elasto-plasticity, Comput. Methods Appl. Mech. Engrg. 51 (1985) 177-208.
33. [33] E. Stein and R. Ahmad, An equilibrium method for stress calculation using finite element displacement models, Comput. Methods Appl. Mech. Engrg. 10 (1977) 175-198.
34. [34] T. Sussman and K.-J. Bathe, A finite element formulation for nonlinear incompressible elastic and inelastic analysis, Comput. Struct. 26 (1987) 357-409.
35. [35] R. Verfürth, A posteriori error estimators for the Stokes equations, Numer. Math. 55 (1989) 309-325.
36. [36] R. Verfürth, A posteriori error estimates for nonlinear problems. Finite element discretizations of elliptic equations, Math. Comput. 62 (1994) 445-475.
37. [37] R. Verfürth, A Review of A Posteriori Error Estimation and Adaptive Mesh-Refinement Techniques (Wiley-Teubner, Chichester, 1996).
38. [38] M.L. Williams, Stress singularities resulting from various boundary conditions in angular corners of plates in extension, J. Appl. Mech. 19 (1952) 526-528.