Blending in the extended finite element method by discontinuous Galerkin and assumed strain methods.

*(English)*Zbl 1195.74175Summary: In the extended finite element method (XFEM), errors are caused by parasitic terms in the approximation space of the blending elements at the edge of the enriched subdomain. A discontinuous Galerkin (DG) formulation is developed, which circumvents this source of error. A patch-based version of the DG formulation is developed, which decomposes the domain into enriched and unenriched subdomains. Continuity between patches is enforced with an internal penalty method. An element-based form is also developed, where each element is considered a patch. The patch-based DG is shown to have similar accuracy to the element-based DG for a given discretization but requires significantly fewer degrees of freedom. The method is applied to material interfaces, cracks and dislocation problems. For the dislocations, a contour integral form of the internal forces that only requires integration over the patch boundaries is developed. A previously developed assumed strain (AS) method is also developed further and compared with the DG method for weak discontinuities and linear elastic cracks. The DG method is shown to be significantly more accurate than the standard XFEM for a given element size and to converge optimally, even where the standard XFEM does not. The accuracy of the DG method is similar to that of the AS method but requires less application-specific coding.

##### Keywords:

discontinuous Galerkin; DG; extended finite element method; XFEM; assumed strain; AS; partition of unity; PU; blending elements; dislocations; cracks; interfaces##### Software:

XFEM
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\textit{R. Gracie} et al., Int. J. Numer. Methods Eng. 74, No. 11, 1645--1669 (2008; Zbl 1195.74175)

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##### References:

[1] | Belytschko, International Journal for Numerical Methods in Engineering 45 pp 601– (1999) |

[2] | Moës, International Journal for Numerical Methods in Engineering 46 pp 131– (1999) |

[3] | Melenk, Computer Methods in Applied Mechanics and Engineering 139 pp 290– (1996) |

[4] | Song, International Journal for Numerical Methods in Engineering 67 pp 868– (2006) |

[5] | Asferg, International Journal for Numerical Methods in Engineering 72 pp 464– (2007) |

[6] | Simone, International Journal for Numerical Methods in Engineering 67 pp 1122– (2006) |

[7] | Ventura, International Journal for Numerical Methods in Engineering 62 pp 1463– (2005) |

[8] | Gracie, International Journal for Numerical Methods in Engineering 69 pp 423– (2007) |

[9] | Sukumar, Computer Methods in Applied Mechanics and Engineering 190 pp 6183– (2001) |

[10] | Chessa, International Journal for Numerical Methods in Engineering 57 pp 1015– (2003) |

[11] | Legay, International Journal for Numerical Methods in Engineering 64 pp 991– (2005) |

[12] | . Triangular mesh methods for the neutron transport equation. Los Alamos Scientific Laboratory Report LA-UR-73-479, 1973. |

[13] | Bassi, Journal of Computational Physics 138 pp 251– (1997) |

[14] | Cockburn, SIAM Journal on Numerical Analysis 35 pp 2440– (1998) |

[15] | Baumann, International Journal for Numerical Methods in Fluids 31 pp 79– (1999) |

[16] | Wheeler, SIAM Journal on Numerical Analysis 15 pp 152– (1978) |

[17] | Arnold, SIAM Journal on Numerical Analysis 19 pp 742– (1982) |

[18] | Arnold, SIAM Journal on Numerical Analysis 39 pp 1749– (2002) |

[19] | Discontinuous Galerkin methods for elastodynamics. Master’s Thesis, Delft University of Technology, 2005. |

[20] | Farhat, Computer Methods in Applied Mechanics and Engineering 190 pp 6455– (2001) |

[21] | Duarte, Computer Methods in Applied Mechanics and Engineering 190 pp 193– (2000) |

[22] | Laborde, International Journal for Numerical Methods in Engineering 64 pp 354– (2005) |

[23] | Farhat, Computer Methods in Applied Mechanics and Engineering 192 pp 1389– (2003) |

[24] | Simo, International Journal for Numerical Methods in Engineering 29 pp 1595– (1990) |

[25] | Stolarski, Computer Methods in Applied Mechanics and Engineering 60 pp 195– (1987) |

[26] | , . Nonlinear Finite Elements for Continua and Structures. Wiley: New York, 2000. · Zbl 0959.74001 |

[27] | Fleming, International Journal for Numerical Methods in Engineering 40 pp 1483– (1997) |

[28] | Stolarska, International Journal for Numerical Methods in Engineering 51 pp 943– (2001) |

[29] | Belytschko, International Journal for Numerical Methods in Engineering 50 pp 993– (2001) |

[30] | Belytschko, International Journal of Plasticity 23 pp 1721– (2007) |

[31] | Gracie, Journal of Mechanics and Physics of Solids (2007) |

[32] | Moran, International Journal of Fracture 35 pp 79– (1987) |

[33] | Liu, International Journal for Numerical Methods in Engineering 59 pp 1103– (2004) |

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.