Strict and effective bounds in goal-oriented error estimation applied to fracture mechanics problems solved with XFEM

International Journal for Numerical Methods in Engineering

Volumn 81, Issue 6, 2010, Pages 671-700

  Panetier, J ^a   Ladeveze P ^a,b   Chamoin, L ^a  

^a Grande Voie des Vignes   (France)

^b EADS Foundation Chair 'Advanced Computational Structural Mechanics'   (France)

Author keywords

Constitutive relation error; Goal oriented error estimation; Strict bounds; XFEM

Indexed keywords

BASIS FUNCTIONS; CONSTITUTIVE RELATION ERRORS; DISCRETIZATION ERRORS; EXTENDED FINITE ELEMENT METHOD; EXTRACTION TECHNIQUES; GOAL-ORIENTED ERROR ESTIMATION; HIGH QUALITY; LEVEL SET; LOCAL ERROR BOUND; MIXED-MODE STRESS; NUMERICAL EXAMPLE; STANDARD FINITE ELEMENT; TECHNICAL ASPECTS;

ACOUSTIC VARIABLES CONTROL; COMPUTER CRIME; ESTIMATION; FRACTURE; FRACTURE MECHANICS; STRESSES;

FINITE ELEMENT METHOD;

EID: 74949087378     PISSN: 00295981     EISSN: 10970207     Source Type: Journal    
DOI: 10.1002/nme.2705     Document Type: Article

References (45)

1. Babuska I, Strouboulis T. The Finite Element Method and its Reliability. Oxford University Press: Oxford, 2001.
2. Stein E (ed.). Error-controlled Adaptive Finite Elements in Solid Mechanics. Wiley: New York, 2003.
3. Ladevèze P, Pelle J-P. Mastering Calculations in Linear and Nonlinear Mechanics. Springer: NY, 2004.
4. Paraschivoiu M, Peraire J, Patera AT. A posteriori finite element bounds for linear functional outputs of elliptic partial differential equations. Computer Methods in Applied Mechanics and Engineering 1997; 150:289-312.
5. Rannacher R, Suttmeier FT. A feedback approach to error control in finite element methods: application to linear elasticity. Computational Mechanics 1997; 19:434-446.
6. Peraire J, Patera AT. Bounds for linear-functional outputs of coercive partial differential equations; local indicators and adaptive refinements. In Advances in Adaptive Computational Methods in Mechanics, Ladeve'ze P, Oden JT (eds). Elsevier: Amsterdam, 1998; 199-216.
7. Cirak F, Ramm E. A posteriori error estimation and adaptivity for linear elasticity using the reciprocal theorem. Computer Methods in Applied Mechanics and Engineering 1998; 156:351-362.
8. Prudhomme S, Oden JT. On goal-oriented error estimation for elliptic problems: application to the control of pointwise errors. Computer Methods in Applied Mechanics and Engineering 1999; 176:313-331.
9. Ladevèze P, Rougeot P, Blanchard P, Moreau JP. Local error estimators for finite element linear analysis. Computer Methods in Applied Mechanics and Engineering 1999; 176:231-246.
10. Becker R, Rannacher R. An optimal control approach to shape a posteriori error estimation in finite element methods. In Acta Numerica, Isereles A (ed.). vol. 10. Cambridge University Press: Cambridge, 2001; 1-120.
11. Ladevèze P. Upper error bounds on calculated outputs of interest for linear and nonlinear structural problems. Comptes Rendus Académie des Sciences-Mécanique, Paris 2006; 334:399-407.
12. Ladevèze P. Strict upper error bounds for calculated outputs of interest in computational structural mechanics. Computational Mechanics 2008; 42(2):271-286.
13. Chamoin L, Ladevèze P. A non-intrusive method for the calculation of strict and efficient bounds of calculated outputs of interest in linear viscoelasticity problems. Computer Methods in Applied Mechanics and Engineering 2008; 197(9-12):994-1014.
14. Panetier J, Ladevèze P, Louf F. Strict bounds for computed stress intensity factors. Computers and Structures 2009; 87(15-16):1015-1021.
15. Stone TJ, Babuska I. A numerical method with a posteriori error estimation for determining the path taken by a propagating crack. Computer Methods in Applied Mechanics and Engineering 1998; 160:245-271.
16. Strouboulis T, Babuska I, Datta D, Copps K, Gangaraj SK. A posteriori estimation and adaptive control of the error in the quantity of interest-part 1: a posteriori estimation of the error in the Von Mises stress and the stress intensity factors. Computer Methods in Applied Mechanics and Engineering 2000; 181:261-294.
17. Heintz P, Samuelson K. On adaptive strategies and error control in fracture mechanics. Computers and Structures 2004; 82:485-497.
18. Rüter M, Stein E. Goal-oriented a posteriori error estimates in linear elastic fracture mechanics. Computer Methods in Applied Mechanics and Engineering 2006; 195:251-278.
19. Xuan ZC, Parès N, Peraire J. Computing upper and lower bounds for the J-integral in two dimensional linear elasticity. Computer Methods in Applied Mechanics and Engineering 2006; 195:430-443.
20. Gallimard L, Panetier J. Error estimation of stress intensity factors for mixed-mode cracks. International Journal for Numerical Methods in Engineering 2006; 68(3):299-316.
21. Xuan ZC, Parès N, Peraire J. Computing lower and upper bounds on stress intensity factors in bimaterials. International Journal for Numerical Methods in Engineering 2007; 42:336-341.
22. Moës N, Dolbow J, Belytschko T. A finite element method for crack growth without remeshing. International Journal for Numerical Methods in Engineering 1999; 46:131-150.
23. Laborde P, Pommier J, Renard Y, Salaun M. High-order extended finite element method for crack domains. International Journal for Numerical Methods in Engineering 2005; 24:354-381.
24. Xiao QZ, Karihaloo BL. Improving the accuracy of XFEM crack tip fields using higher order quadrature and statically admissible stress recovery. International Journal for Numerical Methods in Engineering 2006; 66(9):1378-1410.
25. Strouboulis T, Zhang L, Wang D, Babuska I. A posteriori error estimation for generalized finite element methods. Computer Methods in Applied Mechanics and Engineering 2006; 195:852-879.
26. Bordas S, Duflot M. Derivative recovery and a posteriori error estimate for extended finite elements. Computer Methods in Applied Mechanics and Engineering 2007; 196(35-36):3381-3399.
27. Rodenas JJ, Gonzales-Estrada OA, Tarancon JE, Fuenmayor FJ. A recovery-type error estimator for the extended finite element method based on singular+smooth stress field splitting. International Journal for Numerical Methods in Engineering 2008; 76(4):545-571.
28. Pannachet T, Sluys LJ, Askes H. Error estimation and adaptivity for discontinuous failure. International Journal for Numerical Methods in Engineering 2008; 78(5):528-563.
29. Hild P, Lleras V, Renard Y. A posteriori error analysis for Poisson's equation approximated by XFEM. ESAIM: Proceedings, vol. 27, 2009; 107-121.
30. Bui HD. Fracture Mechanics. Inverse Problems and Solutions. Springer: Berlin, 2006.
31. Barsoum RS. Application of quadratic isoparametric finite elements in linear fracture mechanics. International Journal of Fracture 1974; 10:603-605.
32. Belytschko T, Black T. Elastic crack growth in finite elements with minimal remeshing. International Journal for Numerical Methods in Engineering 1999; 45:601-620.
33. Wyart E, Coulon D, Pardoen T, Remacle JF, Lani F. Application of the substructured finite element/extended finite element method (S-FE/XFE) to the analysis of cracks in aircraft thin walled structures. Engineering Fracture Mechanics 2009; 76(1):44-58.
34. Ladevèze P, Leguillon D. Error estimate procedure in the finite element method and application. SIAM Journal on Numerical Analysis 1983; 20(3):485-509.
35. Stern M, Becker EB, Dunham RS. A contour integral computation of mixed-mode stress intensity factors. International Journal of Fracture 1976; 12:359-368.
36. Babuska I, Miller A. The post-processing approach in the finite element method-part 2: the calculation of stress intensity factors. International Journal for Numerical Methods in Engineering 1984; 20:1111-1129.
37. Leguillon D, Sanchez-Palencia E. Computation of Singular Solutions in Elliptic Problems and Elasticity. Wiley: New York, Masson, Paris, 1987.
38. Ladevèze P, Rougeot P. New advances on a posteriori error on constitutive relation in finite element analysis. Computer Methods in Applied Mechanics and Engineering 1997; 150:239-249.
39. Babuska I, Strouboulis T, Upadhyay CS, Gangaraj SK, Copps K. Validation of a posteriori error estimators by numerical approach. International Journal for Numerical Methods in Engineering 1994; 37:1073-1123.
40. Prudhomme S, Nobile F, Chamoin L, Oden J-T. Analysis of a subdomain-based error estimator for finite element approximations of elliptic problems. Numerical Methods for Partial Differential Equations 2004; 20(2):165-192.
41. Pares N, Diez P, Huerta A. Subdomain-based flux-free a posteriori error estimators. Computer Methods in Applied Mechanics and Engineering 2006; 195(4-6):297-323.
42. Moitinho de Almeida JP, Maunder EAW. Recovery of equilibrium on star patches using a partition of unity technique. Computer Methods in Applied Mechanics and Engineering 2009; DOI: 10.1002/nme.2623.
43. Moës N, Béchet E, Tourbier E. Imposing Dirichlet boundary conditions in the extended finite element method. International Journal for Numerical Methods in Engineering 2006; 67:1641-1669.
44. Chamoin L, Ladevèze P. Bounds on history-dependent or independent local quantities in viscoelasticity problems solved by approximate methods. International Journal for Numerical Methods in Engineering 2007; 71(12): 1387-1411.
45. Chamoin L, Ladevèze P. Strict and practical bounds through a non-intrusive and goal-oriented error estimation method for linear viscoelasticity problems. Finite Elements in Analysis and Design 2009; 45:251-262.