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Computational Mechanics

, Volume 54, Issue 2, pp 523–533 | Cite as

A stabilization technique for the regularization of nearly singular extended finite elements

  • Stefan Loehnert
Original Paper

Abstract

In this contribution a simple, robust and efficient stabilization technique for extended finite element (XFEM) simulations is presented. It is useful for arbitrary crack geometries in two or three dimensions that may lead to very bad condition numbers of the global stiffness matrix or even ill-conditioning of the equation system. The method is based on an eigenvalue decomposition of the element stiffness matrix of elements that only possess enriched nodes. Physically meaningful zero eigenmodes as well as enrichment scheme dependent numerically reasonable zero eigenmodes are filtered out. The remaining subspace is stabilized depending on the magnitude of the respective eigenvalues. One of the main advantages is the fact that neither the equation solvers need to be changed nor the solution method is restricted. The efficiency and robustness of the method is demonstrated in numerous examples for 2D and 3D fracture mechanics.

Keywords

XFEM Stabilization Cracks Conditioning 

Notes

Acknowledgments

The support of the German Research Foundation (DFG) within the framework of the grant SFB 871 is gratefully acknowledged.

References

  1. 1.
    Moës N, Dolbow J, Belytschko T (1999) A finite element method for crack growth without remeshing. Int J Numer Meth Eng. 46:131–150CrossRefzbMATHGoogle Scholar
  2. 2.
    Belytschko T, Black T (1999) Elastic crack growth in finite elements with minimal remeshing. Int J Numer Meth Eng 45:601–620MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Strouboulis T, Copps K, Babuška I (2001) The generalized finite element method. Comput Methods Appl Mech Eng. 190:4081–4193CrossRefzbMATHGoogle Scholar
  4. 4.
    Melenk J, Babuška I (1996) The partition of unity finite element method: basic theory and applications. Comput Methods Appl Mech Eng 139:289–314CrossRefzbMATHGoogle Scholar
  5. 5.
    Fries T, Belytschko T (2010) The extended/generalized finite element method: an overview of the method and its applications. Int J Numer Meth Eng 84:253–304Google Scholar
  6. 6.
    Sukumar N, Moës N, Moran B, Belytschko T (2000) Extended finite element method for three-dimensional crack modelling. Int J Numer Meth Eng 48:1549–1570CrossRefzbMATHGoogle Scholar
  7. 7.
    Strouboulis T, Babuška I, Copps K (2000) The design and analysis of the generalized finite element method. Comput Methods Appl Mech Eng 181:43–69CrossRefzbMATHGoogle Scholar
  8. 8.
    Béchet E, Minnebo H, Moës N, Burgardt B (2005) Improved implementation and robustness study of the x-fem for stress analysis around cracks. Int J Numer Meth Eng 64:1033–1056CrossRefzbMATHGoogle Scholar
  9. 9.
    Menk A, Bordas S (2011) A robust preconditioning technique for the extended finite element method. Int J Numer Meth Eng 85:1609–1632MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Farhat C, Roux F (1991) A method of finite element tearing and interconnecting and its parallel solution algorithm. Int J Numer Meth Eng 32:1205–1227 CrossRefzbMATHGoogle Scholar
  11. 11.
    Babuška I, Banerjee U (2012) Stable generalized finite element method (sgbem). Comput Methods Appl Mech Eng 201–204:91–111CrossRefGoogle Scholar
  12. 12.
    Fries TP, Baydoun M, Weber N (2013) 3d crack propagation with the xfem and a hybrid explicit-implicit crack description. In: Gravouil A, Renard Y, Combescure A (eds) International Conference on Extended Finite Element Methods XFEM 2013Google Scholar
  13. 13.
    Reusken A (2008) Analysis of an extended pressure finite element space for two-phase incompressible flows. Comput Visual Sci 11:293–305MathSciNetCrossRefGoogle Scholar
  14. 14.
    Wriggers P, Korelc J (1996) On enhanced strain methods for small and finite deformation solids. Comput Mech 18:413–428CrossRefzbMATHGoogle Scholar
  15. 15.
    Wriggers P (2008) Nonlinear Finite Element Methods. Springer, BerlinzbMATHGoogle Scholar
  16. 16.
    Laborde P, Pommier J, Renard Y, Salaün M (2005) High-order extended finite element method for cracked domains. Int J Numer Meth Eng 64:354–381CrossRefzbMATHGoogle Scholar
  17. 17.
    Fries T (2008) A corrected xfem approximation without problems in blending elements. Int J Numer Meth Eng 75:503–532MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Loehnert S, Mueller-Hoeppe D, Wriggers P (2011) 3d corrected xfem approach and extension to finite deformation theory. Int J Numer Meth Eng 86:431–452MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Bittencourt T, Wawrzynek P, Ingraffea A, Sousa J (1996) Quasi-automatic simulation of crack propagation for 2d lefm problems. Eng Fract Mech 55:321–334CrossRefGoogle Scholar
  20. 20.
    Ingraffea A, Grigoriu M (1990) Probabilistic fracture mechanics: A validation of predictive capability. Tech. Rep. 90–8, Department of Structural Engineering, Cornell UniversityGoogle Scholar
  21. 21.
    Bocca P, Carpinteri A, Valente S (1991) Mixed mode fracture of concrete. Int J Solids Structures 27:1139–1153CrossRefGoogle Scholar
  22. 22.
    Moës N, Belytschko T (2002) Extended finite element method for cohesive crack growth. Eng Fract Mech 69:813–833CrossRefGoogle Scholar
  23. 23.
    Moës N, Cloirec M, Cartraud P, Remacle JF (2003) A computational approach to handle complex microstructure geometries. Comput Methods Appl Mech Eng 192:3163–3177CrossRefzbMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2014

Authors and Affiliations

  1. 1.Institute of Continuum MechanicsHannoverGermany

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