Vol. 15 No. 2 (2016): Revista UIS Ingenierías
Articles

Comparison of 3-D Integration Schemes for Enriched Elements in XFEM

Vicente Francisco Gonzalez Albuixech
Universidad Politécnica de Valencia
Eugenio Giner Maravilla
Universidad Politécnica de Valencia
Bio
Jose Enrique Tarancon
Universidad Politécnica de Valencia
Portada RUI 15.2

Published 2016-06-16

Keywords

  • enriched elements, fracture mechanics, integration, XFEM.,
  • enriched elements,
  • fracture mechanics,
  • integration,
  • XFEM

How to Cite

Gonzalez Albuixech, V. F., Maravilla, E. G., & Tarancon, J. E. (2016). Comparison of 3-D Integration Schemes for Enriched Elements in XFEM. Revista UIS Ingenierías, 15(2), 7–16. https://doi.org/10.18273/revuin.v15n2-2016001

Abstract

The XFEM is a technique developed for the numerical simulation of fracture mechanics problems. The method shows several advantages. But some questions arise that should be handled with some care. Numerical integration of the enriched elements is one of these issues. In this work we have compared two avalaible techniques for its integration. One is a classic integration scheme and another is a scheme specifically developed for this kind of elements. The differences on the results are minimal, but not in their implementation difficulty. Hence, the classical integration is recommended.

Downloads

Download data is not yet available.

References

  1. R. Barsoum, “Triangular quarter-point elements as elastic and perfectly plastic crack tip elements,” International Journal of Numerical Methods in Engineering, vol. 11, núm. 1, pp. 85–98, 1977. DOI: 10.1002/nme.1620110109
  2. R. Gallagher, “A review of finite element techniques in fracture mechanics,” lst Conference on Numerical Methods in Fracture Mechanics, (Swansea, R.U.), 1978, pp. 1–25.
  3. A.J. Fawkes, D.R.J. Owen y A.R. Luxmoore. “An assessment of crack tip singularity models for use with isoparametric elements,” Engineering Fracture Mechanics, vol. 11, núm. 1, pp. 143–159, 1979. DOI:10.1016/0013-7944(79)90035-3
  4. Y.W. Kwon y J.E. Akin, “Development of a derivative singular element for application to crack propagation problems,” Computers & Structures, vol. 31, núm. 3, pp. 467–471, 1989.
  5. G. Strang, G.J. Fix, An Analysis of the Finite Element Method, Upper Saddle River, EEUU: Prentice-Hall, 1973.
  6. T. Belytschko y T. Black, “Elastic crack growth in finite elements with minimal remeshing,” International Journal of Numerical Methods in Engineering, vol. 45, núm. 5, pp. 601–620, 1999. DOI: 10.1002/(SICI)1097-0207(19990620)45:5<601::AID-NME598>3.0.CO;2-S
  7. M. Fleming, et al. “Enriched element-free galerkin methods for crack-tip field.” International Journal of Numerical Methods in engineering, vol. 40, núm. 8, pp. 1483–1504, abr. 1997.
  8. I. Babuška, G. Caloz, J. Osborn, “Special finite element methods for a class of second order elliptic problems with rough coefficients,” SIAM Journal on Numerical Analysis, vol. 31, núm. 4, pp. 945–981, ago. 1994. http://www.jstor.org/stable/2158110
  9. J.M. Melenk e I. Babuška, “The partition of unity finite element method: Basic theory and applications,” Computer Methods in Applied Mechanics and Engineering, vol. 39, núm. 1–4, pp. 289–314, dic. 1996. DOI: 10.1016/S0045-7825(96)01087-0.
  10. I. Babuška y J. M. Melenk, “The partition of unity method,” International Journal of Numerical methods in engineering, vol. 40, núm. 4 pp. 727–758, 1997. DOI: 10.1002/(SICI)1097-0207
  11. (19970228)40:4<727::AID-NME86>3.0.CO;2-N.
  12. N. Moës, J. Dolbow y T. Belytschko, “A finite element method for crack growth without remeshing,” International Journal for Numerical Methods in Engineering, vol. 46, núm. 1, pp. 131–150, 1999. DOI: 10.1002/(SICI)1097-0207(19990910)46:1<131::AID-NME726>3.0.CO;2-J.
  13. C. Daux et al. “Arbitrary branched and intersecting cracks with the extended finite element method,” International Journal of Numerical Methods in Engineering, vol. 48, núm. 12, pp. 1741–1760, ago. 2000. DOI: 10.1002/1097-0207(20000830)48:12<1741::AID-NME956>3.0.CO;2-L.
  14. N. Moës, A. Gravouil y T. Belytschko, “Non-planar 3d crack growth by the extended finite element and level sets-Part I: Mechanical model,” International Journal for Numerical Methods in Engineering, vol. 53, núm. 11, pp. 2549–2568, abr. 2002. DOI: 10.1002/nme.429.
  15. A. Gravouil, N. Moës y T. Belytschko, “Non-planar 3D crack growth by the extended finite element and level sets—Part II: Level set update,” International Journal for Numerical Methods in Engineering, vol. 53, núm. 11, pp. 2569–2586, abr. 2002. DOI: 10.1002/nme.430.
  16. N. Sukumar et al., “Extended finite element method for three-dimensional crack modelling,” International Journal of Numerical Methods in Engineering, vol. 48, núm. 11, pp. 1549–1570, 2000. DOI: 10.1002/1097-0207(20000820)48:11<1549::AID-NME955>3.0.CO;2-A.
  17. F. L.Stazi, E. Budyn, J. Chessa et al. “An extended finite element method with higher-order elements for curved cracks,” Computational Mechanics, vol. 31, núm. 1, pp. 38–48, may. 2003. DOI: 10.1007/s00466-002-0391-2.
  18. J. Chessa, H. Wang y T. Belytschko, “On the construction of blending elements for local partition of unity enriched finite elements,” International Journal for Numerical Methods in Engineering, vol. 57, núm. 7, pp. 1015–1038, jun. 2003. DOI: 10.1002/nme.777.
  19. T.P. Fries, “A corrected XFEM approximation without problems in blending elements,” International Journal for Numerical Methods in Engineering, vol. 75, núm. 5, pp. 503-532, jul. 2008. DOI: 10.1002/nme.2259.
  20. J.E. Tarancón et al., “Enhanced blending elements for XFEM applied to linear elastic fracture mechanics,” International Journal for Numerical Methods in Engineering, vol. 77, núm. 1, pp. 126–148, ene. 2009. DOI: 10.1002/nme.2402.
  21. E. Béchet et al., “Improved implementation and robustness study of the X-FEM for stress analysis around cracks,” International Journal of Numerical Methods in Engineering, vol. 64, núm. 8, pp. 1033–1056, oct. 2005. DOI: 10.1002/nme.1386.
  22. P. Laborde et al., “High-order extended finite element method for cracked domains,” International Journal of Numerical Methods in Engineering, vol. 64, núm. 3, pp. 354–381, sep. 2005. DOI: 10.1002/nme.1370.
  23. N. Sukumar y J. H. Prevost, “Modeling quasi-static crack growth with the extended finite element method part I: Computer implementation,” International Journal of Solids and Structures, vol. 40, núm. 26, pp. 7513–7537, dic. 2003. DOI: 10.1016/j.ijsolstr.2003.08.002
  24. B.L. Karihaloo y Q.Z. Xiao, “Modelling of stationary and growing cracks in FE framework without remeshing: a state-of-the-art review,” Computers & Structures, vol. 81, núm. 3, pp. 119–129, feb. 2003. DOI: 10.1016/S0045-7949(02)00431-5.
  25. Q. Xiao y B. Karihaloo, “Improving the accuracy of XFEM crack tip fields using higher order quadrature and statically admissible stress recovery,” International Journal of Numerical Methods in Engineering, vol. 66, núm. 9, pp. 1378–1410, may. 2006. DOI: 10.1002/nme.1601.
  26. K. Park et al., “Integration of singular enrichment functions in the generalized/extended finite element method for three-dimensional problems,” International Journal for Numerical Methods in Engineering, vol. 78, núm.10, pp. 1220–1257, jun. 2009. DOI: 10.1002/nme.2530.
  27. J. Dolbow, N. Moës y T. Belytschko, “An extended finite element method for modelling crack growth with frictional contact,” Computer Methods in Applied Mechanics and Engineering, vol. 190, núm. 51-52, pp. 6825–6846, oct. 2001. DOI: 10.1016/S0045-7825(01)00260-2.
  28. J. Chessa, H. Wang y T. Belytschko, “On the construction of blending elements for local partition of unity enriched finite elements,” International Journal for Numerical Methods in Engineering, vol. 57, núm. 7, pp. 1015–1038, jun. 2003. DOI: 10.1002/nme.777.
  29. A. Quarteroni, R. Sacco y F. Saleri, Numerical mathematics, Berlin, Alemanía: Springer, 2007.
  30. T. Strouboulis, I Babuška y K. Copps, “The design and analysis of the generalized finite element method,” Computer Methods in Applied Mechanics and Engineering, vol. 181, núm. 1-3, pp. 43–69, ene. 2000. DOI: 10.1016/S0045-7825(99)00072-9.
  31. T. Strouboulis, K. Copps y I. Babuška, “The generalized finite element method: an example of its implementation and illustration of its performance,” International Journal of Numerical Methods in Engineering, vol. 47, núm. 8, pp. 1401–1417, mar. 2000. DOI: 10.1002/(SICI)1097-0207(20000320)47:8<1401::AID-NME835>3.0.CO;2-8.
  32. C. Daux et al., “Arbitrary branched and intersecting cracks with the extended finite element method,” International Journal of Numerical Methods in Engineering, vol. 48, núm.12, pp. 1741–1760, ago. 2000. DOI: 10.1002/1097-0207(20000830)48:12<1741::AID-NME956>3.0.CO;2-L.
  33. Element partitioning code in 2-D and 3-D for the extended finite element method. 2016. Disponible en: http://dilbert.engr.ucdavis.edu/~suku/xfem
  34. A.H. Stroud, Approximate Calculation of Multiple Integrals, Upper Saddle River, EEUU: Prentice-Hall, 1971.
  35. P. Keast, “Moderate-degree tetrahedral quadrature formulas,” Computer Methods in Applied Mechanics and Engineering, vol. 55, núm. 3, pp. 339–348, may. 1986. DOI:10.1016/0045-7825(86)90059-9.
  36. G. Ventura, “On the elimination of quadrature subcells for discontinuous functions in the extended finite-element method,” International Journal of Numerical Methods in Engineering, vol. 66, núm. 5, pp. 761–795, abr. 2006. DOI: 10.1002/nme.1570