Strain smoothing in FEM and XFEM

Computers and Structures

Volumn 88, Issue 23-24, 2010, Pages 1419-1443

  Bordas, Stéphane P A ^a   Rabczuk, Timon ^b   Hung, Nguyen Xuan ^c   Nguyen, Vinh Phu ^d   Natarajan, Sundararajan ^e   Bog, Tino ^b   Quan, Do Minh ^c   Hiep, Nguyen Vinh ^c  

^a UNIVERSITY OF GLASGOW   (United Kingdom)

^b UNIVERSITY OF CANTERBURY   (New Zealand)

^c UNIVERSITY OF NATURAL SCIENCES   (Viet Nam)

^d DELFT UNIVERSITY OF TECHNOLOGY   (Netherlands)

^e GE AVIATION   (United States)

Author keywords

Accuracy; Convergence; Extended finite element method; SFEM; Smoothed finite element; XFEM

Indexed keywords

ACCURACY; CONVERGENCE; EXTENDED FINITE ELEMENT METHOD; SFEM; SMOOTHED FINITE ELEMENTS; XFEM;

CONVERGENCE OF NUMERICAL METHODS; INTEGRATION;

FINITE ELEMENT METHOD;

EID: 77958104869     PISSN: 00457949     EISSN: None     Source Type: Journal    
DOI: 10.1016/j.compstruc.2008.07.006     Document Type: Conference Paper

References (109)

1. ABAQUS. Standard user's manual,version 6.4. Rawtucket, RI: Hibbitt, Karlsson and Sorensen Inc.; 2004.
2. J.P. Moitinho De Almeida, and O.J.B. Almeida Pereira A set of hybrid equilibrium finite element models for the analysis of three-dimensional solids Int J Numer Method Eng 39 1996 2789 2802
3. M. Arroyo, and M. Ortiz Local maximum-entropy approximation schemes Lect Notes Comput Sci Eng 57 2006 1
4. M. Arroyo, and M. Ortiz Local maximum-entropy approximation schemes: a seamless bridge between finite elements and meshfree methods Int J Numer Method Eng 65 13 2006 2167 2202
5. I. Babuška, and I. Melenk Partition of unity method Int J Numer Method Eng 40 4 1997 727 758
6. F.B. Barros, S.P.B. Proena, and C.S. de Barcellos On error estimator and p-adaptivity in the generalized finite element method Int J Numer Method Eng 60 14 2004 2373 2398
7. K.J. Bathe Finite element procedures 1996 Prentice-Hall, Masetchuset (MIT) Englewood Cliffs, NJ
8. K.J. Bathe, and E.N. Dvorkin A four-node plate bending element based on Mindlin/Reissener plate theory and a mixed interpolation Int J Numer Method Eng 21 1985 367 383
9. K.J. Bathe, and E.N. Dvorkin A formulation of general shell elements. the use of mixed interpolation of tensorial components Int J Numer Method Eng 22 1986 697 722
10. E. Béchet, H. Minnebo, N. Moës, and B. Burgardt Improved implementation and robustness study of the X-FEM for stress analysis around cracks Int J Numer Method Eng 64 8 2005 1033 1056
11. T. Belytschko, and T. Black Elastic crack growth in finite elements with minimal remeshing Int J Numer Method Eng 45 1999 601 620
12. T. Belytschko, Y.Y. Lu, and L. Gu Element-free Galerkin methods Int J Numer Method Eng 37 1994 229 256
13. T. Belytschko, N. Moës, S. Usui, and C. Parimi Arbitrary discontinuities in finite elements Int J Numer Method Eng 50 4 2001 993 1013
14. I. Leviathan, and T. Belytschko Physical stabilization of the 4-node shell element with one-point quadrature Comput Method Appl Mech Eng 113 1994 321 350
15. H. Boduroglu, and F. Erdogan Internal and edge cracks in a plate of finite width under bending J Appl Mech 50 1983 621 628
16. Bordas S, Duflot M. Derivative recovery and a posteriori error estimation in extended finite element methods. Comput Method Appl Mech Eng 2007;196(35-36):3381-9. .
17. Bordas S, Legay A. Enriched finite element short course: class notes. In: The extended finite element method, a new approach to numerical analysis in mechanics: course notes. Organized by S. Bordas and A. Legay through the EPFL school of continuing education, Lausanne, Switzerland, December 7-9, 2005 and July 2007.
18. Bordas S, Moran B. Enriched finite elements and level sets for damage tolerance assessment of complex structures. Eng Fract Mech 2006;73(9):1176-201, doi:10.1016/j.engfracmech.2006.01.006.
19. Bordas SPA, Conley JG, Moran B, Gray J, Nichols E. A simulation-based design paradigm for complex cast components. Eng Comput 2007;23(1):25-37. doi:10.1007/s00366-006-0030-1.
20. Bordas S, Duflot M, Le P. A simple error estimator for extended finite elements. Commun Numer Method Eng 2008;24(11), 961-71. doi:10.1002/cnm.1001.
21. Bordas S, Nguyen VP, Dunant C, Hung A, Guidoum Nguyen-Dang. An extended finite element library. Int J Numer Method Eng 2007;71(6):703-32. doi:10.1002/nme.1966. (Pubitemid 47226130)
22. J.S. Chen, C.T. Wu, S. Yoon, and Y. You A stabilized conforming nodal integration for Galerkin mesh-free methods Int J Numer Method Eng 50 2001 435 466
23. Chessa J. The extended finite element method for free surface and two phase flow problems. PhD thesis, Northwestern University; 2002.
24. Chessa J, Belytschko T. An enriched finite element method and level sets for axisymmetric two-phase flow with surface tension. Int J Numer Method Eng 2003;58(13):2041-64. doi:10.1002/nme.946. (Pubitemid 37529403)
25. J. Chessa, and T. Belytschko The extended finite element method for two-phase fluids ASME J Appl Mech 70 1 2003 10 17
26. J. Chessa, P. Smolinski, and T. Belytschko The extended finite element method (X-FEM) for solidification problems Int J Numer Method Eng 53 2002 1957 1977
27. S. Commend, and Th. Zimmermann Object-oriented nonlinear finite element programming: a primer Adv Eng Softw 32 8 2001 611 628
28. S. Commend, A. Truty, and Th. Zimmermann Stabilized finite elements applied to elastoplasticity: I. Mixed displacement pressure formulation Comput Method Appl Mech Eng 193 2004 3559 3586
29. R.D. Cook Improved two-dimensional finite element J Struct Div ASCE 100 1974
30. K.Y. Dai, G.R. Liu, and T.T. Nguyen An n-sided polygonal smoothed finite element method (nSFEM) for solid mechanics Finite Element Anal Des 43 11 2007 847 860
31. J. Dolbow, and A. Devan Enrichment of enhanced assumed strain approximations for representing strong discontinuities: addressing volumetric incompressibility and the discontinuous patch test Int J Numer Method Eng 59 1 2004 47 67
32. J. Dolbow, N. Moës, and T. Belytschko Modeling fracture in Mindlin-Reissner plates with the extended finite element method Int J Solid Struct 37 2000 7161 7183
33. Duddu R, Bordas S, Chopp D, Moran B. A combined extended finite element and level set method for biofilm growth. Int J Numer Method Eng 2007;74(5):848-70. doi:10.1002/nme.2200.
34. Duflot M, Bordas S. A posteriori error estimation for extended finite elements by an extended global recovery. Int J Numer Method Eng 2008;76(8):1123-38. doi:10.1002/nme.2332.
35. C. Dunant, P. Nguyen, M. Belgasmia, S. Bordas, A. Guidoum, and Nguyen-Dang Hung Architecture trade-offs of including a mesher in an object-oriented extended finite element code Eur J Comput Mech 16 2007 237 258 (special issue on the extended finite element method)
36. E.N. Dvorkin, and K.J. Bathe A continuum mechanics based four-node shell element for general nonlinear analysis Eng Comput 1 1994 77 88
37. B. Moran, M. Fleming, Y.A. Chu, and T. Belytschko Enriched element free Galerkin methods for singular fields Int J Numer Method Eng 40 1997 1483 1504
38. Fries T-P, Matthies HG. Classification and overview of meshfree methods. Informatirkbericht Report 2003-03, Institute of Scientific Computing, Technical University Braunschweig, Hans-Sommer-Strasse 65, D38106 Braunschweig; 2003.
39. Fries TP. A corrected XFEM approximation without problems in blending elements. Int J Numer Method Eng 2008;75(5):503-32. doi:10.1002/nme.2259.
40. O. Garcia, E.A. Fancello, C.S. Barcellos, and C.A. Duarte Hp clouds in Mindlin's thick plate model Int J Numer Method Eng 47 2000 1381 1400
41. R. Hill The mathematical theory of plasticity 1950 Oxford University Press
42. W.K. Liu, and T.J.R. Hughes Nonlinear finite element analysis of shells. Part II: Two-dimensional shells Comput Method Appl Mech Eng 27 1981 167 182
43. Hung Nguyen-Xuan, Bordas Stéphane, Hung Nguyen-Dang. Collection of papers from Prof. Nguyen-Dang Hung former students, chapter Smooth strain finite elements: selective integration. Vietnam National University-Ho Chi Minh City Publishing House; 2006. Géry de Saxcé and Nicolas Moës.
44. Huynh DBP, Belytschko T. The extended finite element method for fracture in composite materials. Int J Numer Method Eng 2008;77(2):214-39. doi:10.1002/nme.2411.
45. J. Chessa, H. Wang, and T. Belytschko On the construction of blending elements for local partition of unity enriched finite elements Int J Numer Method Eng 57 2003 1015 1038
46. J.L. Batoz, and M.B. Tahar Evaluation of a new quadrilateral thin plate bending element Int J Numer Method Eng 18 1982 1655 1677
47. P. Krysl, and T. Belytschko Analysis of thin plates by the element-free Galerkin method Comput Mech 16 1995 1 10
48. P. Krysl, and T. Belytschko Analysis of thin shells by the element-free Galerkin method Int J Numer Method Eng 33 1996 3057 3080 (Pubitemid 126377013)
49. P. Laborde, J. Pommier, Y. Renard, and M. Salaun High order extended finite element method for cracked domains Int J Numer Method Eng 190 47 2004 6183 6200
50. Legrain G, Moës N, Huerta A. Stability of incompressible formulations enriched with X-FEM. Comput Method Appl Mech Eng 2008;197(21-24):1835-49. doi:10.1016/j.cma.2007.08.032.
51. Q. Li, J. Soric, T. Jarak, and S.N. Atluri A locking-free meshless local Petrov-Galerkin formulation for thick and thin plates J Comput Phys 208 2005 116 133
52. G.R. Liu, T.T. Nguyen, K.Y. Dai, and K.Y. Lam Theoretical aspects of the smoothed finite element method (SFEM) Int J Numer Method Eng 71 8 2007 902 930
53. Liu GR, Dai KY, Nguyen TT. A smoothed finite element for mechanics problems. Comput Mech 2007;30(6):859-77. doi:10.1007/s00466-006-0075-4. (Pubitemid 46327302)
54. D.S. Malkus, and T.J.R. Hughes Mixed finite element methods - reduced and selective integration technique: a unification of concepts Comput Method Appl Mech Eng 15 1978 63 81
55. Melenk JM. On generalized finite element methods. PhD thesis, University of Maryland, College Park, MD; 1995.
56. J.M. Melenk, and I. Babuška The partition of unity finite element method: basic theory and applications Comput Method Appl Mech Eng 139 1996 289 314
57. N. Moës, J. Dolbow, and T. Belytschko A finite element method for crack growth without remeshing Int J Numer Method Eng 46 1 1999 131 150
58. B. Moran, and C.F. Shih Crack tip and associated domain integrals from momentum and energy balance Eng Fract Mech 27 1987 615 641
59. Natarajan Sundararajan, Bordas Stéphane. Smoothed polygonal FEM based on Laplace and Wachspress interpolants. Int J Numer Method Eng, submitted for publication.
60. Sundararajan Natarajan and Stéphane Bordas. Smoothed extended polygonal FEM. Int J Numer Method Eng, submitted for publication.
61. Neittaanmaki P, Rossi T, Korotov S, Onate E, Periaux J, Knorzer D. Polygonal interpolants: construction and adaptive computations on quadtree meshes.
62. T.T. Nguyen, G.R. Liu, K.Y. Dai, and K.Y. Lam Selective smoothed finite element method Tsinghua Sci Technol 12 5 2007 497 508
63. Nguyen VP, Rabczuk T, Bordas S. Meshless methods: A review and computer implementation aspects. Math Comput Simul 2008;79(3):763-813. doi:10.1016/j.matcom.2008.01.003.
64. Nguyen-Thanh N, Rabczuk T, Nguyen-Xuan H, Bordas SPA. A smoothed finite element method for shell analysis. Comput Method Appl Mech Eng 2008;198(2):165-77. doi:10.1016/j.cma.2008.05.029.
65. Nguyen-Xuan Hung, S. Bordas, and Nguyen-Dang Hung Smooth finite elements: convergence, accuracy and properties Int J Numer Method Eng 74 2 2008 175 208 10.1002/nme.2146
66. Hung Nguyen-Xuan, Bordas S, Hung Nguyen-Dang. Smooth finite elements: connecting displacement and equilibrium finite elements, applications to elasto-plasticity. Comput Method Appl Mech Eng, submitted for publication.
67. Nguyen-Xuan Hung, T. Rabczuk, S. Bordas, and J.-F. Debongnie A smoothed finite element method for plate analysis Comput Method Appl Mech Eng 197 13-16 2008 1184 1203 10.1016/j.cma.2007.10.008
68. D.R.J. Owen, and E. Hinton Finite elements in plasticity - theory and practice 1980 Pineridge Press Limited Swansea, UK
69. Pannachet T, Díez P, Askes H. Error estimation and p-adaptivity based on the partition of unity method. In: ICCST '02; 2002. p. 39-40 [ISBN: 0-948749-81-4].
70. T. Rabczuk, and P.M.A. Areias A meshfree thin shell for arbitrary evolving cracks based on an external enrichment Comput Model Eng Sci 16 2 2006 115 130
71. T. Rabczuk, Stéphane Bordas, and G. Zi A three-dimensional meshfree method for continuous crack initiation, nucleation and propagation in statics and dynamics Comput Mech 40 3 2007 473 495 10.1007/s00466-006-0122-1
72. T. Rabczuk, G. Zi, S. Bordas, and Nguyen-Xuan Hung A geometrically nonlinear three-dimensional cohesive crack method for reinforced concrete structures Eng Fract Mech 75 16 2008 4740 4758 10.1016/j.engfracmech.2008.06.019
73. T. Rabczuk, P.M.A. Areias, and T. Belytschko A meshfree thin shell method for non-linear dynamic fracture Int J Numer Method Eng 72 5 2007 524 548
74. J.R. Rice A path independent integral and the approximate analysis of strain concentration by notches and cracks J Appl Mech 1968 379 386
75. L.F. Richardson The approximate arithmetical solution by finite differences of physical problems Trans R Soc Lond A210 1910 307 357
76. E.P. Kasper, and R.L. Taylor A mixed-enhanced strain method Comput Struct 75 2000 237 250
77. Rodenas JJ, Gonzalez-Estrada OA, Tarancon JE, Fuenmayor FJ. A recovery-type error estimator for the extended finite element method based on singular + smooth stress field splitting. Int J Numer Method Eng 2008;76(4):545-71. doi:10.1002/nme.2313.
78. M. Rüter, and E. Stein Adaptive finite element analysis of crack propagation in elastic fracture mechanics based on averaging techniques Comput Mater Sci 31 3-4 2004 247 257
79. M. Rüter, and E. Stein Goal-oriented a posteriori error estimates in linear elastic fracture mechanics Comput Methods Appl Mech Eng 195 2006 251 278
80. M. Rüter, and E. Stein On the duality of finite element discretization error control in computational Newtonian and Eshelbian mechanics Comput Mech 39 5 2007 609 630
81. M. Rüter, S. Korotov, and C. Steenbock Goal-oriented error estimates based on different FE-spaces for the primal and the dual problem with applications to fracture mechanics Comput Mech 39 6 2007 787 797
82. J.C. Simo, and T.J.R. Hughes Computational inelasticity 1999 Springer-Verlag Berlin
83. M.S. Rifai, J.C. Simo, and D.D. Fox On a stress resultant geometrically exact shell model. Part II: The linear theory; computational aspects Comput Method Appl Mech Eng 73 1989 53 92
84. J. Soric, Q. Li, T. Jarak, and S.N. Atluri The meshless local Petrov-Galerkin (MLPG) formulation for analysis of thick plates Comput Model Eng Sci 6 2004 349 357
85. T. Strouboulis, I. Babuška, and K. Copps The design and analysis of the generalized finite element method Comput Method Appl Mech Eng 181 2000 43 69
86. T. Strouboulis, K. Copps, and I. Babuška The generalized finite element method : an example of its implementation and illustration of its performance Int J Numer Method Eng 47 8 2000 1401 1417
87. T. Strouboulis, L. Zhang, D. Wang, and I. Babuška A posteriori error estimation for generalized finite element methods Comput Method Appl Mech Eng 195 9-12 2006 852 879
88. N. Sukumar Construction of polygonal interpolants: a maximum entropy approach Int J Numer Method Eng 61 2004 2159 2181
89. N. Sukumar Maximum entropy approximation Bayesian Inference and Maximum Entropy Methods in Science and Engineering 803 2005 337 344
90. N. Sukumar, and A. Tabarraei Conforming polygonal finite elements Int J Numer Method Eng 61 2004 2045 2066
91. Sukumar N, Tabarraei A. Numerical formulation and application of polygonal finite elements. In: Proceedings of the seventh international ESAFORM conference on material forming, Trondheim, Norway; 2004. p. 4-28.
92. N. Sukumar, and R.W. Wright Overview and construction of meshfree basis functions: from moving least squares to entropy approximants Int J Numer Method Eng 70 2007 181 205
93. N. Sukumar, B. Moran, and T. Belytschko The natural element method in solid mechanics Int J Numer Method Eng 43 5 1998 839 887
94. Tabarraei A, Sukumar N. Extended finite element method on polygonal and quadtree meshes. Comput Method Appl Mech Eng 2008;197(5):425-38.
95. Stéphane Bordas, T. Rabczuk, and G. Zi Three-dimensional crack initiation, propagation, branching and junction in non-linear materials by an extended meshfree method without asymptotic enrichment Eng Fract Mech 75 5 2008 943 960
96. G. Ventura On the elimination of quadrature subcells for discontinuous functions in the extended finite-element method Int J Numer Method Eng 66 5 2006 761 795
97. Ventura Giulio, Gracie Robert, Belytschko Ted. Fast integration and weight function blending in the extended finite element method. Int J Numer Method Eng 2008;77(1):1-29. doi:10.1002/nme.2387.
98. D. Wang, and J.S. Chen Locking-free stabilized conforming nodal integration for meshfree Mindlin-Reissner plate formulation Comput Method Appl Mech Eng 193 2004 1065 1083
99. S.S. Wang, J.F. Yau, and H.T. Corten A mixed-mode crack analysis of rectilinear anisotropic solids using conservation laws of elasticity Int J Fract 16 3 1980 247 259
100. S.L. Weissman, and R.L. Taylor Resultant fields for mixed plate bending elements Comput Method Appl Mech Eng 79 1990 321 355
101. H.M. Westergaard Bearing and cracks J Appl Mech 61 1939 49 53
102. Wyart E. Three-dimensional crack analysis in aeronautical structures using the substructured finite element/extended finite element method. PhD thesis, Université catholique de Louvain; 2007.
103. Wyart E, Coulon D, Duflot M, Pardoen T, Remacle J-F, Lani F. A substructured FE-shell/XFE-3D method for crack analysis in thin-walled structures. Int J Numer Method Eng 2007;72(7):757-79. doi:10.1002/nme.2029. (Pubitemid 47618155)
104. Wyart E, Coulon D, Martiny P, Pardoen T, Remacle J-F, Lani F. A substructured FE/XFE method for stress intensity factors computation in an industrial structure. Rev Eur Mech Numer, in press.
105. Wyart E, Duflot M, Coulon D, Martiny P, Pardoen T, Remacle J-F, et al. Substructuring FE-XFE approaches applied to three-dimensional crack propagation. J Comput Appl Math 2008;215(2):626-39. doi:10.1016/j.cam.2006.03.066. (Pubitemid 351400889)
106. B.Z. Xiao, and B.L. Karihaloo Improving the accuracy of XFEM crack tip fields using higher order quadrature and statically admissible stress recovery Int J Numer Method Eng 66 9 2006 1378 1410
107. J.W. Yoo, B. Moran, and J.S. Chen Stabilized conforming nodal integration in the natural-element method Int J Numer Method Eng 60 2004 861 890
108. O.C. Zienkiewicz, and R.L. Taylor The finite element method 5th ed. 2000 Butterworth-Heinemann Oxford
109. M. Zlamal Superconvergence and reduced integration in the finite element method Math Comput 32 143 1978 663 685