Computational Mechanics

, Volume 44, Issue 1, pp 73–92 | Cite as

Generalized finite element method enrichment functions for curved singularities in 3D fracture mechanics problems

  • J. P. Pereira
  • C. A. Duarte
  • X. Jiao
  • D. Guoy
Original Paper


This paper presents a study of generalized enrichment functions for 3D curved crack fronts. Two coordinate systems used in the definition of singular curved crack front enrichment functions are analyzed. In the first one, a set of Cartesian coordinate systems defined along the crack front is used. In the second case, the geometry of the crack front is approximated by a set of curvilinear coordinate systems. A description of the computation of derivatives of enrichment functions and curvilinear base vectors is presented. The coordinate systems are automatically defined using geometrical information provided by an explicit representation of the crack surface. A detailed procedure to accurately evaluate the surface normal, conormal and tangent vectors along curvilinear crack fronts in explicit crack surface representations is also presented. An accurate and robust definition of orthonormal vectors along crack fronts is crucial for the proper definition of enrichment functions. Numerical experiments illustrate the accuracy and robustness of the proposed approaches.


Partition of unity methods Generalized/Extended finite element method 3D fracture mechanics Crack front enrichments 


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  1. 1.
    Babuška I, Melenk JM (1997) The partition of unity finite element method. Int J Numer Methods Eng 40: 727–758zbMATHCrossRefGoogle Scholar
  2. 2.
    Belytschko T, Black T (1999) Elastic crack growth in finite elements with minimal remeshing. Int J Numer Methods Eng 45: 601–620zbMATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    Chopp DL, Sukumar N (2003) Fatigue crack propagation of multiple coplanar cracks with the coupled extended finite element/fast marching method. Int J Eng Sci 41: 845–869CrossRefMathSciNetGoogle Scholar
  4. 4.
    Duarte CA, Babuška I, Oden JT (2000) Generalized finite element methods for three dimensional structural mechanics problems. Comp Struct 77: 215–232CrossRefGoogle Scholar
  5. 5.
    Duarte CA, Hamzeh ON, Liszka TJ, Tworzydlo WW (2001) A generalized finite element method for the simulation of three-dimensional dynamic crack propagation. Comp Methods Appl Mech Eng 190(15–17):2227–2262.
  6. 6.
    Duarte CA, Reno LG, Simone A (2007) A high-order generalized FEM for through-the-thickness branched cracks. Int J Numer Methods Eng 72(3):325–351. Google Scholar
  7. 7.
    Duarte CAM, Oden JT (1996) An hp adaptive method using clouds. Comp Methods Appl Mech Eng 139: 237–262zbMATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    Duflot M (2007) A study of the representation of cracks with level sets. Int J Numer Methods Eng 70: 1261–1302CrossRefMathSciNetGoogle Scholar
  9. 9.
    Fleming M, Chu YA, Moran B, Belytschko T (1997) Enriched element-free Galerkin methods for crack tip fields. Int J Numer Methods Eng 40: 1483–1504CrossRefMathSciNetGoogle Scholar
  10. 10.
    Jiao X (2007) Face offsetting: a unified framework for explicit moving interfaces. J Comput Phys 220(2): 612–625zbMATHCrossRefMathSciNetGoogle Scholar
  11. 11.
    Jiao X, Bayyana NR, Zha H (2007) Optimizing surface triangulation via near isometry with reference meshes. In: Geert YS, van Albada D, Dongarra J, Sloot PMA (eds) Computational science—ICCS 2007, pp 334–341, Beijing, China, May. Proceedings, Part I, Springer, HeidelbergGoogle Scholar
  12. 12.
    Keast P (1986) Moderate-degree tetrahedral quadrature formulas. Comp Methods Appl Mech Eng 55: 339–348zbMATHCrossRefMathSciNetGoogle Scholar
  13. 13.
    Lancaster P, Salkauskas K (1981) Surfaces generated by moving least squares methods. Math Comput 37(155): 141–158zbMATHCrossRefMathSciNetGoogle Scholar
  14. 14.
    Lancaster P, Salkauskas K (1986) Curve and surface fitting, an introduction. Academic Press, San DiegozbMATHGoogle Scholar
  15. 15.
    Lebedev LP, Cloud MJ (2003) Tensor analysis. World Scientific, New JerseyzbMATHGoogle Scholar
  16. 16.
    Melenk JM, Babuška I (1996) The partition of unity finite element method: basic theory and applications. Comp Methods Appl Mech Eng 139: 289–314zbMATHCrossRefGoogle Scholar
  17. 17.
    Mergheim J, Kuhl E, Steinmann P (2005) A finite element method for the computational modeling of cohesive cracks. Int J Numer Methods Eng 63: 276–289zbMATHCrossRefGoogle Scholar
  18. 18.
    Moës N, Belytschko T (2002) Extended finite element method for cohesive crack growth. Eng Fracture Mech 69: 813–833CrossRefGoogle Scholar
  19. 19.
    Moës N, Dolbow J, Belytschko T (1999) A finite element method for crack growth without remeshing. Int J Numer Methods Eng 46: 131–150zbMATHCrossRefGoogle Scholar
  20. 20.
    Moës N, Gravouil A, Belytschko T (2002) Non-planar 3D crack growth by the extended finite element and level sets—part I: mechanical model. Int J Numer Methods Eng 53(11): 2549–2568zbMATHCrossRefGoogle Scholar
  21. 21.
    Murakami Y (1992) Stress intensity factors handbook, vol 3, 1st edn. Pergamon, OxfordGoogle Scholar
  22. 22.
    Oden JT, Duarte CA (1997) Chapter: clouds, cracks and FEM’s. In: Reddy BD (ed) Recent developments in computational and applied mechanics, pp 302–321, Barcelona, Spain. International Center for Numerical Methods in Engineering, CIMNE (1997)Google Scholar
  23. 23.
    Oden JT, Duarte CA, Zienkiewicz OC (1998) A new cloud-based hp finite element method. Comp Methods Appl Mech Eng 153: 117–126zbMATHCrossRefMathSciNetGoogle Scholar
  24. 24.
    Oden JT, Duarte CAM (1997) Chapter: solution of singular problems using hp clouds. In: Whiteman JR (eds) The mathematics of finite elements and applications—highlights 1996. Wiley, New York, pp 35–54Google Scholar
  25. 25.
    Park K, Pereira JP, Duarte CA, Paulino GH (2008) Integration of singular enrichment functions in the generalized/extended finite element method for three-dimensional problems. Int J Numer Methods Eng.
  26. 26.
    Pereira JP, Duarte CA (2004) Computation of stress intensity factors for pressurized cracks using the generalized finite element method and superconvergent extraction techniques. In: Lyra PRM, da Silva SMBA, Magnani FS, do N. Guimaraes LJ, da Costa LM, Parente E Jr (eds) XXV Iberian Latin-American congress on computational methods in engineering. Recife, PE, Brazil, November. 15 pp. ISBN Proceedings CD: 857 409 869-8 (2004)Google Scholar
  27. 27.
    Pereira JP, Duarte CA (2005) Extraction of stress intensity factors from generalized finite element solutions. Eng Anal Bound Elem 29: 397–413CrossRefGoogle Scholar
  28. 28.
    Pereira JP, Duarte CA, Guoy D, Jiao X (2008) Hp-generalized FEM and crack surface representation for non-planar 3D cracks. Int J Numer Meth Eng.
  29. 29.
    Raju JC, Newman IS Jr (1979) Stress-intensity factors for a wide range of semi-elliptical surface cracks in finite-thickness plates. Eng Fracture Mech 11: 817–829CrossRefGoogle Scholar
  30. 30.
    Reddy JN, Rasmussen ML (1982) Advanced engineering analysis. Wiley, New YorkGoogle Scholar
  31. 31.
    Simone A (2004) Partition of unity-based discontinuous elements for interface phenomena: computational issues. Commun Numer Methods Eng 20: 465–478zbMATHCrossRefGoogle Scholar
  32. 32.
    Simone A, Wells GN, Sluys LJ (2003) From continuous to discontinuous failure in a gradient-enhanced continuum damage model. Comp Methods Appl Mech Eng 192(41–42): 4581–4607zbMATHCrossRefGoogle Scholar
  33. 33.
    Strouboulis T, Copps K, Babuška I (2001) The generalized finite element method. Comp Methods Appl Mech Eng 190: 4081–4193zbMATHCrossRefGoogle Scholar
  34. 34.
    Sukumar N, Chopp DL, Moran B (2003) Extended finite element method and fast marching method for three-dimensional fatigue crack propagation. Eng Fracture Mech 70: 29–48CrossRefGoogle Scholar
  35. 35.
    Sukumar N, Moës N, Moran B, Belytschko T (2000) Extended finite element method for three-dimensional crack modelling. Int J Numer Methods Eng 48(11): 1549–1570zbMATHCrossRefGoogle Scholar
  36. 36.
    Szabo B, Babuška I (1991) Finite element analysis. Wiley, New YorkzbMATHGoogle Scholar
  37. 37.
    Szabo BA, Babuška I (1988) Computation of the amplitude of stress singular terms for cracks and reentrant corners. In: Cruse TA (ed) Fracture mechanics: nineteenth symposium. ASTM STP 969, pp 101–124, Southwest Research Institute, San Antonio (1988)Google Scholar
  38. 38.
    Tada H, Paris P, Irwin G (2000) The stress analysis of cracks handbook, 3rd edn. ASME Press, New YorkGoogle Scholar
  39. 39.
    Walters MC, Paulino GH, Dodds RH Jr (2004) Stress-intensity factors for surface cracks in functionally graded materials under mode-I thermomechanical loading. Int J Solids Struct 41: 1081–1118zbMATHCrossRefGoogle Scholar
  40. 40.
    Wells GN, de Borst R, Sluys LJ (2002) A consistent geometrically non-linear approach for delamination. Int J Numer Methods Eng 54: 1333–1355zbMATHCrossRefGoogle Scholar
  41. 41.
    Wells GN, Sluys LJ (2001) A new method for modeling cohesive cracks using finite elements. Int J Numer Methods Eng 50: 2667–2682zbMATHCrossRefGoogle Scholar
  42. 42.
    Zi G, Belytschko T (2003) New crack-tip elements for XFEM and applications to cohesive cracks. Int J Numer Methods Eng 57: 2221–2240zbMATHCrossRefGoogle Scholar

Copyright information

© Springer-Verlag 2008

Authors and Affiliations

  • J. P. Pereira
    • 1
  • C. A. Duarte
    • 1
  • X. Jiao
    • 2
  • D. Guoy
    • 3
  1. 1.Department of Civil and Environmental EngineeringUniversity of Illinois at Urbana-ChampaignUrbanaUSA
  2. 2.Department of Applied Mathematics and StatisticsStony Brook UniversityStony BrookUSA
  3. 3.Center for Simulation of Advanced RocketsUniversity of Illinois at Urbana-ChampaignUrbanaUSA

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