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Advances in Discretization Methods pp 209–228Cite as

Stabilized X-FEM for Heaviside and Nonlinear Enrichments

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Part of the book series: SEMA SIMAI Springer Series ((SEMA SIMAI,volume 12))

Abstract

One of the drawbacks of the eXtended Finite Element Method and similar approaches, like the Generalized Finite Element Method, is the problem of ill-conditioning of the related systems of equations at the solution stage. This occurs for example in Heaviside function enrichments when the discontinuity is close to discretisation nodes but also for non-linear enrichment functions used in conjunction to geometric enrichment domains. In the present work the motivation of ill-conditioning is analyzed to derive a novel methodology for stabilization, based on setting proper constraints for the variables. This methodology does not impact on the initial formulation nor in the element stiffness computation, so that it is very effective for engineering applications. Results are analyzed in 1D and 3D to show its performance and properties.

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References

  1. ABAQUS: ABAQUS Documentation. Dassault Systèmes, Providence (2014)

    Google Scholar 

  2. Abdelaziz, Y., Hamouine, A.: A survey of the extended finite element. Comput. Struct. 86, 1141–1151 (2008)

    Article  Google Scholar 

  3. Babuška, I., Banerjee, U.: Stable generalized finite element method (SGFEM). Comput. Methods Appl. Mech. Eng. 201–204, 91–111 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  4. Babuška, I., Banerjee, U., Osborn, J.: Survey of meshless and generalized finite element methods: a unified approach. Acta Numer. 12, 1–125 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  5. Béchet, E., Minnebo, H., Moës, N., Burgardt, B.: Improved implementation and robustness study of the X-FEM for stress analysis around cracks. Int. J. Numer. Methods Eng. 64, 1033–1056 (2005)

    Article  MATH  Google Scholar 

  6. Belytschko, T., Gracie, R., Ventura, G.: A review of extended/generalized finite element methods for material modeling. Model. Simul. Mater. Sci. Eng. 17, p. Article number 043001 (2009)

    Google Scholar 

  7. Benvenuti, E.: Mesh–size–objective XFEM for regularized continuous–discontinuous transition. Finite Elem. Anal. Des. 47, 1326–1336 (2011)

    Article  Google Scholar 

  8. Benvenuti, E.: XFEM with equivalent eigenstrain for matrix–inclusion interfaces. Comput. Mech. 53, 893–908 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  9. Benvenuti, E., Tralli, A.: Simulation of finite-width process zone for concrete-like materials. Comput. Mech. 50, 479–497 (2012)

    Article  MATH  Google Scholar 

  10. Benvenuti, E., Tralli, A., Ventura, G.: A regularized XFEM model for the transition from continuous to discontinuous displacements. Int. J. Numer. Methods Eng. 74, 911–944 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  11. Benvenuti, E., Ventura, G., Ponara, N., Tralli, A.: Variationally consistent extended FE model for 3D planar and curved imperfect interfaces. Comput. Methods Appl. Math. Eng. 267, 1–22 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  12. Berger-Vergiat, L., Waisman, H., Hiriyur, B., Tuminaro, R., Keyes, D.: Inexact Schwarz-algebraic multigrid preconditioners for crack problems modeled by extended finite element methods. Int. J. Numer. Methods Eng. 90, 311–328 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  13. Duarte, C., Kim, D.: Analysis and applications of a generalized finite element method with global–local enrichment functions. Comput. Methods Appl. Mech. Eng. 197, 487–504 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  14. Ferté, G., Massin, P., Moës, N.: Convergence analysis of linear or quadratic X-FEM for curved free boundaries. Comput. Methods Appl. Mech. Eng. 278, 794–827 (2014)

    Article  MathSciNet  Google Scholar 

  15. Fries, T.: A corrected XFEM approximation without problems in blending elements. Int. J. Numer. Methods Eng. 75, 503–532 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  16. Fries, T.P., Belytschko, T.: The extended/generalized finite element method: an overview of the method and its applications. Int. J. Numer. Methods Eng. 84, 253–304 (2010)

    MathSciNet  MATH  Google Scholar 

  17. Geuzaine, C., Remacle, J.-F.: Gmsh: A 3-d finite element mesh generator with built-in pre-and post-processing facilities. Int. J. Numer. Methods Eng. 79, 1309–1331 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  18. Gupta, V., Duarte, C.A., Babuška, I., Banerjee, U.: Stable GFEM (SGFEM): improved conditioning and accuracy of GFEM/XFEM for three-dimensional fracture mechanics. Comput. Methods Appl. Mech. Eng. 289, 355–386 (2015)

    Article  MathSciNet  Google Scholar 

  19. Koehler, J.: On the dislocation theory of plastic deformation. Phys. Rev. 60, 397–410 (1941)

    Article  Google Scholar 

  20. Laborde, P., Pommier, J., Renard, Y., Salaun, M.: High order extended finite element method for cracked domains. Int. J. Numer. Meth. Eng. 64, 354–381 (2005)

    Article  MATH  Google Scholar 

  21. Lang, C., Makhija, D., Doostan, A., Maute, K.: A simple and efficient preconditioning scheme for heaviside enriched XFEM. Numer. Anal. arXiv:1312.6092v1 (2013)

    Google Scholar 

  22. Loehnert, S.: A stabilization technique for the regularization of nearly singular extended finite elements. Comput. Mech. 54, 523–533 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  23. LS-DYNA: User’s Manual. Livermore Software Technology Corporation, Livermore (2013)

    Google Scholar 

  24. Martin, A., Esnault, J.-B., Massin, P.: About the use of standard integration schemes for X-FEM in solid mechanics plasticity. Comput. Methods Appl. Mech. Eng. 283, 551–572 (2015):

    Article  MathSciNet  Google Scholar 

  25. Menk, A., Bordas, S.P.A.: A robust preconditioning technique for the extended finite element method. Int. J. Numer. Methods Eng. 85, 1609–1632 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  26. Mousavi, S.E., Pask, J.E., Sukumar, N.: Efficient adaptive integration of functions with sharp gradients and cusps in n-dimensional parallelepipeds. Int. J. Numer. Methods Eng. 91, 343–357 (2012)

    Article  MATH  Google Scholar 

  27. Müller, B., Kummer, F., Oberlack, M.: Highly accurate surface and volume integration on implicit domains by means of moment-fitting. Int. J. Numer. Methods Eng. 96, 512–528 (2013)

    Article  MathSciNet  Google Scholar 

  28. Nabarro, F.: Theory of Crystal Dislocations. Dover Publications, New York (1967)

    Google Scholar 

  29. Sillem, A., Simone, A., Sluys, L.J.: The orthonormalized generalized finite element method-OGFEM: efficient and stable reduction of approximation errors through multiple orthonormalized enriched basis functions. Comput. Methods Appl. Mech. Eng. 287, 112–149 (2015)

    Article  MathSciNet  Google Scholar 

  30. Strouboulis, T., Babuška, I., Copps, K.: The design and analysis of the generalized finite element method. Comput. Methods Appl. Mech. Eng. 181, 43–69 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  31. Sudhakar, Y., Wall, W.A.: Quadrature schemes for arbitrary convex/concave volumes and integration of weak form in enriched partition of unity methods. Comput. Methods Appl. Mech. Eng. 258, 39–54 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  32. Ventura, G.: On the elimination of quadrature subcells for discontinuous functions in the extended finite-element method. Int. J. Numer. Methods Eng. 66, 761–795 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  33. Ventura, G., Moran, B., Belytschko, T.: Dislocations by partition of unity. J. Numer. Methods Eng. 62, 1463–1487 (2005)

    Article  MATH  Google Scholar 

  34. Ventura, G., Gracie, R., Belytschko, T.: Fast integration and weight function blending in the extended finite element method. Int. J. Numer. Methods Eng. 77, 1–29 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  35. Ventura, G., Benvenuti, E.: Equivalent polynomials for quadrature in heaviside function enriched elements. Int. J. Numer. Methods Eng. 102, 688–710 (2015)

    Article  MathSciNet  Google Scholar 

  36. Waisman, H., Berger-Vergiat, L.: An adaptive domain decomposition preconditioner for crack propagation problems modeled by XFEM. Int. J. Multiscale Comput. Eng. 11, 633–654 (2013)

    Article  Google Scholar 

  37. Yazid, A., Abdelkader, N., Abdelmadjid, H.: A state-of-the-art review of the X-FEM for computational fracture mechanics. Appl. Math. Model. 33, 4269–4282 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  38. Zhang, Q., Banerjee, U., Babuška, I.: Higher order stable generalized finite element method. Numer. Math. 128, 1–29 (2014)

    Article  MathSciNet  MATH  Google Scholar 

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Authors and Affiliations

  1. DISEG, Politecnico di Torino, Corso Duca degli Abruzzi 24, Torino, Italy

    Giulio Ventura & Claudia Tesei

Authors
  1. Giulio Ventura
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  2. Claudia Tesei
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Correspondence to Giulio Ventura .

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Editors and Affiliations

  1. Structural, Building, Geotechnical Eng., Politecnico di Torino, Torino, Italy

    Giulio Ventura

  2. Department of Engineering, University of Ferrara, Ferrara, Italy

    Elena Benvenuti

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Ventura, G., Tesei, C. (2016). Stabilized X-FEM for Heaviside and Nonlinear Enrichments. In: Ventura, G., Benvenuti, E. (eds) Advances in Discretization Methods. SEMA SIMAI Springer Series, vol 12. Springer, Cham. https://doi.org/10.1007/978-3-319-41246-7_10

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