Higher order tip enrichment of extended finite element method in thermoelasticity.

*(English)*Zbl 1344.74062Summary: An eXtended Finite Element Method (XFEM) is presented that can accurately predict the stress intensity factors (SIFs) for thermoelastic cracks. The method uses higher order terms of the thermoelastic asymptotic crack tip fields to enrich the approximation space of the temperature and displacement fields in the vicinity of crack tips – away from the crack tip the step function is used. It is shown that improved accuracy is obtained by using the higher order crack tip enrichments and that the benefit of including such terms is greater for thermoelastic problems than for either purely elastic or steady state heat transfer problems. The computation of SIFs directly from the XFEM degrees of freedom and using the interaction integral is studied. Directly computed SIFs are shown to be significantly less accurate than those computed using the interaction integral. Furthermore, the numerical examples suggest that the directly computed SIFs do not converge to the exact SIFs values, but converge roughly to values near the exact result. Numerical simulations of straight cracks show that with the higher order enrichment scheme, the energy norm converges monotonically with increasing number of asymptotic enrichment terms and with decreasing element size. For curved crack there is no further increase in accuracy when more than four asymptotic enrichment terms are used and the numerical simulations indicate that the SIFs obtained directly from the XFEM degrees of freedom are inaccurate, while those obtained using the interaction integral remain accurate for small integration domains. It is recommended in general that at least four higher order terms of the asymptotic solution be used to enrich the temperature and displacement fields near the crack tips and that the \(J\)- or interaction integral should always be used to compute the SIFs.

##### MSC:

74S05 | Finite element methods applied to problems in solid mechanics |

74F05 | Thermal effects in solid mechanics |

76R99 | Diffusion and convection |

PDF
BibTeX
XML
Cite

\textit{A. Zamani} et al., Comput. Mech. 46, No. 6, 851--866 (2010; Zbl 1344.74062)

Full Text:
DOI

**OpenURL**

##### References:

[1] | Areias PMA, Belytschko T (2007) Two-scale method for shear bands: thermal effects and variable bandwidth. Int J Numer Methods Eng 72: 658–696 · Zbl 1194.74355 |

[2] | Barrett R et al (1998) Templates for the solution of linear systems: building blocks for iterative methods. Soci Ind Appl Math. Available at http://www.siam.org/books |

[3] | 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 Methods Eng 64: 1033–1056 · Zbl 1122.74499 |

[4] | Belytschko T, Black T (1999) Elastic crack growth in finite elements with minimal remeshing. Int J Numer Methods Eng 45: 601–620 · Zbl 0943.74061 |

[5] | Belytschko T, Moës N, Usui S, Parimi C (2001) Arbitrary discontinuities in finite elements. Int J Numer Methods Eng 50: 993–1013 · Zbl 0981.74062 |

[6] | Belytschko T, Gracie R, Ventura G (2009) A review of extended/generalized finite element methods for material modeling. Modelling Simul Mater Sci Eng 17(4) · Zbl 1195.74201 |

[7] | Bordas S, Duflot M (2007) Derivative recovery and a posteriori error estimate for extended finite elements. Comput Methods Appl Mech Eng 196: 3381–3399 · Zbl 1173.74401 |

[8] | Chen YZ, Hasebe N (2003) Solution for a curvilinear crack in a thermoelastic medium. J Thermal Stresses 26: 245–259 |

[9] | Dongarra J, Lumsdaine A, Pozo R, Remington K (1998) IML++ version 1.2: Iterative methods library reference guide. National Institute of Standards and Technology, University of Notre Dame, Available at http://math.nist.gov/iml++ |

[10] | Duarte C, Reno L, Simone A (2007) A high-order generalized fem for through-the-thickness branched cracks. Int J Numer Meth Eng 72: 325–351 · Zbl 1194.74385 |

[11] | Duarte CA, Hamzeh ON, Liszka TJ, Tworzydlo WW (2001) A generalized finite element method for the simulation of three dimensional dynamic crack propagation. Comput Methods Appl Mech Eng 190: 2227–2262 · Zbl 1047.74056 |

[12] | Duflot M (2008) The extended finite element method in thermoelastic fracture mechanics. Int J Numer Methods Eng 74: 827–847 · Zbl 1195.74170 |

[13] | Duflot M, Bordas S (2008) A posteriori error estimation for extended finite elements by an extended global recovery. Int J Numer Methods Eng 76: 1123–1138 · Zbl 1195.74171 |

[14] | Fries TP (2008) A corrected XFEM approximation without problems in blending elements. Int J Numer Methods Eng 75: 503–532 · Zbl 1195.74173 |

[15] | Gracie R, Wang H, Belytschko T (2008) Blending in the extended finite element method by discontinuous Galerkin and assumed strain methods. Int J Numer Methods Eng 74: 1645–1669 · Zbl 1195.74175 |

[16] | Hansbo A, Hansbo P (2004) A finite element method for the simulation of strong and weak discontinuities in solid mechanics. Comput Methods Appl Mech Eng 193: 3523–3540 · Zbl 1068.74076 |

[17] | Hetnarski RB, Eslami MR (2009) Thermal stresses–advanced theory and applications, solid mechanics and its applications, vol 158. Springer, Berlin |

[18] | Huang R, Sukumar N, Prévost JH (2003) Modeling quasi-static crack growth with the extended finite element method. Part II. Numerical applications. Int J Solids Struct 40: 7539–7552 · Zbl 1064.74163 |

[19] | Karihaloo BL, Xiao QZ (2003) Modelling of stationary and growing cracks in fe framework without remeshing: a state-of-the-art review. Comput Struct 81: 119–129 |

[20] | Laborde P, Pommier J, Renard Y, Salaün M (2005) High-order extended finite element method for cracked domains. Int J Numer Methods Eng 64: 354–381 · Zbl 1181.74136 |

[21] | Liu X, Xiao Q, Karihaloo BL (2004) XFEM for direct evaluation of mixed mode sifs in homogeneous and bi-materials. Int J Numer Methods Eng 59: 1103–1118 · Zbl 1041.74543 |

[22] | Melenk JM, Babuška I (1996) The partition of unity fnite element method: basic theory and applications. Comput Methods Appl Mech Eng 139: 289–314 · Zbl 0881.65099 |

[23] | Moës N, Belytschko T (2002) Extended finite element method for cohesive crack growth. Eng Fract Mech 69(2): 813–833 |

[24] | Moës N, Dolbow J, Belytschko T (1999) A finite element method for crack growth without remeshing. Int J Numer Methods Eng 46: 131–150 · Zbl 0955.74066 |

[25] | Moran B, Shih CF (1987a) Crack tip and associated domain integrals from momentum and energy balance. Eng Fract Mech 27: 615–641 |

[26] | Moran B, Shih CF (1987b) A general treatment of crack tip contour integrals. Int J Fract 35: 295–310 |

[27] | Mousavi SE, Xiao H, Sukumar N (2009) Generalized gaussian quadrature rules on arbitrary polygons. Int J Numer Methods Eng. doi: 10.1002/nme.2759 · Zbl 1183.65026 |

[28] | Natarajan S, Bordas S, Mahapatra DR (2009) Numerical integration over arbitrary polygonal domains based on Schwarz–Christoffel conformal mapping. Int J Numer Methods Eng 80: 103–134 · Zbl 1176.74190 |

[29] | Pozo R, Remington K, Lumsdaine A (1998) SparseLib++ version 1.7: Sparse Matrix Library. National Institute of Standards and Technology, University of Notre Dame. Available at http://math.nist.gov/sparselib++ |

[30] | Rabczuk T, Bordas S, Zi G (2010) On three-dimensional modelling of crack growth using partition of unity methods. Comput Struct (in press) |

[31] | Réthore J, Roux S, Hild F (2010) Hybrid analytical and extended finite element method (HAX-FEM): a new enrichment procedure for cracked solids. Int J Numer Methods Eng 81: 269–285 · Zbl 1183.74309 |

[32] | Shih CF, Moran B, Nakamura T (1986) Energy release rate along a three-dimensional crack front in a thermally stressed body. Int J Fract 30: 79–102 |

[33] | Song JH, Areias PMA, Belytschko T (2006) A method for dynamic crack and shear band propagatin with phantom nodes. Int J Numer Methods Eng 67: 868–893 · Zbl 1113.74078 |

[34] | Stazi F, Budyn E, Chessa J, Belytschko T (2003) An extended finite element method with higher-order elements for curved cracks. Comput Mech 31: 38–48 · Zbl 1038.74651 |

[35] | Sukumar N (2000) Element partitioning code in 2-d and 3-d for the extended finite element method, available from http://dilbert.engr.ucdavis.edu/suku/xfem |

[36] | Ventura G (2006) On the elimination of quadrature subcells for discontinuous functions in the extended finite element method. Int J Numer Methods Eng 66: 761–795 · Zbl 1110.74858 |

[37] | Ventura G, Gracie R, Belytschko T (2009) Fast integration and weight function blending in the extended finite element method. Int J Numer Methods Eng 77: 1–29 · Zbl 1195.74201 |

[38] | Williams ML (1957) One the stress distribution at the base of a stationary crack. J Appl Mech 24: 109–114 · Zbl 0077.37902 |

[39] | Wilson WK (1969) Combined mode fracture mechanics. PhD thesis, University of Pittsburgh |

[40] | Yau J, Wang S, Corten H (1980) A mixed-mode crack analysis of isotropic solids using conservation laws of elasticity. J Appl Mech 47: 335–341 · Zbl 0463.73103 |

[41] | Yosibash Z (1996) Numerical thermo-elastic analysis of singularities in two-dimensions. Int J Fract 74: 341–361 |

[42] | Zamani A, Eslami MR (2009) Coupled dynamical thermoelasticity of a functionally graded cracked layer. J Thermal Stresses 32: 969–985 |

[43] | Zamani A, Eslami MR (2010) Implementation of the extended finite element method for dynamic thermoelastic fracture initiation. Int J Solids Struct 47: 1392–1404 · Zbl 1193.74161 |

This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.