Higher-order XFEM for curved strong and weak discontinuities.

*(English)*Zbl 1188.74052Summary: The extended finite element method (XFEM) enables the accurate approximation of solutions with jumps or kinks within elements. Optimal convergence rates have frequently been achieved for linear elements and piecewise planar interfaces. Higher-order convergence for arbitrary curved interfaces relies on two major issues: (i) an accurate quadrature of the Galerkin weak form for the cut elements and (ii) a careful formulation of the enrichment, which should preclude any problems in the blending elements. For (i), we employ a strategy of subdividing the elements into subcells with only one curved side. Reference elements that are higher-order on only one side are then used to map the integration points to the real element. For (ii), we find that enrichments for strong discontinuities are easily extended to higher-order accuracy. In contrast, problems in blending elements may hinder optimal convergence for weak discontinuities. Different formulations are investigated, including the corrected XFEM. Numerical results for several test cases involving strong or weak curved discontinuities are presented. Quadratic and cubic approximations are investigated. Optimal convergence rates are achieved using the standard XFEM for the case of a strong discontinuity. Close-to-optimal convergence rates for the case of a weak discontinuity are achieved using the corrected XFEM.

##### MSC:

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

76M10 | Finite element methods applied to problems in fluid mechanics |

##### Software:

XFEM
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\textit{K. W. Cheng} and \textit{T.-P. Fries}, Int. J. Numer. Methods Eng. 82, No. 5, 564--590 (2010; Zbl 1188.74052)

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##### References:

[1] | Beckett, A moving mesh finite element method for the solution of two-dimensional stefan problems, Journal of Computational Physics 168 (2) pp 500– (2002) |

[2] | Beckett, Computational solution of two-dimensional unsteady pdes using moving mesh methods, Journal of Computational Physics 182 (2) pp 478– (2002) · Zbl 1016.65062 |

[3] | Wawrzynek, An interactive approach to local remeshing around a propagating crack, Finite Elements in Analysis and Design 5 pp 87– (1989) |

[4] | Bouchard, Numerical modelling of crack propagation: automatic remeshing and comparison of different criteria, Computer Methods in Applied Mechanics and Engineering 192 pp 3887– (2003) · Zbl 1054.74724 |

[5] | MoÃ“s, A finite element method for crack growth without remeshing, International Journal for Numerical Methods in Engineering 46 pp 131– (1999) · Zbl 0955.74066 |

[6] | Belytschko, Arbitrary discontinuities in finite elements, International Journal for Numerical Methods in Engineering 50 pp 993– (2001) · Zbl 0981.74062 |

[7] | BabusÌka, The partition of unity method, International Journal for Numerical Methods in Engineering 40 pp 727– (1997) |

[8] | Melenk, The partition of unity finite element method: basic theory and applications, Computer Methods in Applied Mechanics and Engineering 139 pp 289– (1996) · Zbl 0881.65099 |

[9] | Strouboulis, The design and analysis of the generalized finite element method, Computer Methods in Applied Mechanics and Engineering 181 pp 43– (2000) · Zbl 0983.65127 |

[10] | Strouboulis, The generalized finite element method: an example of its implementation and illustration of its performance, International Journal for Numerical Methods in Engineering 47 pp 1401– (2000) · Zbl 0955.65080 |

[11] | Strouboulis, The generalized finite element method, Computer Methods in Applied Mechanics and Engineering 190 pp 4081– (2001) · Zbl 0997.74069 |

[12] | Barros, On error estimator and p-adaptivity in the generalized finite element method, International Journal for Numerical Methods in Engineering 60 pp 2373– (2004) · Zbl 1075.74634 |

[13] | Barros, Generalized finite element method in structural nonlinear analysis-a p-adaptive strategy, Computational Mechanics 33 pp 95– (2004) · Zbl 1067.74063 |

[14] | Duarte, H-p cloudsâan h-p meshless method, Numerical Methods for Partial Differential Equations 12 pp 673– (1996) |

[15] | Duarte, Generalized finite element methods for three-dimensional structural mechanics problems, Computers and Structures 77 pp 215– (2000) |

[16] | Oden, A new cloud-based hp finite element method, Computer Methods in Applied Mechanics and Engineering 153 pp 117– (1998) · Zbl 0956.74062 |

[17] | Legay, Strong and weak arbitrary discontinuities in spectral finite elements, International Journal for Numerical Methods in Engineering 64 pp 991– (2005) · Zbl 1167.74045 |

[18] | Osher, Level Set Methods and Dynamic Implicit Surfaces (2003) |

[19] | Osher, Level set methods: an overview and some recent results, Journal of Computational Physics 169 pp 463– (2001) · Zbl 0988.65093 |

[20] | Sethian, Level Set Methods and Fast Marching Methods (1999) · Zbl 0929.65066 |

[21] | Sukumar, Extended finite element method for three-dimensional crack modelling, International Journal for Numerical Methods in Engineering 48 pp 1549– (2000) · Zbl 0963.74067 |

[22] | Stazi, An extended finite element method with higher-order elements for curved cracks, Computational Mechanics 31 pp 38– (2003) · Zbl 1038.74651 |

[23] | Zi, New crack-tip elements for XFEM and applications to cohesive cracks, International Journal for Numerical Methods in Engineering 57 pp 2221– (2003) · Zbl 1062.74633 |

[24] | DrÃ©au K, Chevaugeon N, MoÃ“s N. High order extended finite element method: influence of the geometrical representation. Proceedings of the 8th World Congress on Computational Mechanics (WCCM VIII), Venice, Italy, 2008. |

[25] | Ventura, On the elimination of quadrature subcells for discontinuous functions in the eXtended finite element method, International Journal for Numerical Methods in Engineering 66 pp 761– (2006) · Zbl 1110.74858 |

[26] | Ventura, Fast integration and weight function blending in the extended finite element method, International Journal for Numerical Methods in Engineering 77 pp 1– (2009) · Zbl 1195.74201 |

[27] | Lee, Combined extended and superimposed finite element method for cracks, International Journal for Numerical Methods in Engineering 59 pp 1119– (2004) · Zbl 1041.74542 |

[28] | Bordas, Strain smoothing in fem and xfem, Computers and Structures (2008) |

[29] | Bordas S, Natarajan S, Duflot M, Nguyen-Xuan H, Rabczuk T. The smoothed extended finite element method. Proceedings of the 8th World Congress on Computational Mechanics (WCCM VIII), Venice, Italy, 2008. |

[30] | Chessa, On the construction of blending elements for local partition of unity enriched finite elements, International Journal for Numerical Methods in Engineering 57 pp 1015– (2003) · Zbl 1035.65122 |

[31] | Fries, The intrinsic partition of unity method, Computational Mechanics 40 pp 803– (2007) · Zbl 1162.74049 |

[32] | Fries, A corrected XFEM approximation without problems in blending elements, International Journal for Numerical Methods in Engineering 75 pp 503– (2008) · Zbl 1195.74173 |

[33] | Chahine, Crack tip enrichment in the XFEM using a cutoff function, International Journal for Numerical Methods in Engineering 75 pp 629– (2008) · Zbl 1195.74167 |

[34] | Laborde, High-order extended finite element method for cracked domains, International Journal for Numerical Methods in Engineering 64 pp 354– (2005) · Zbl 1181.74136 |

[35] | Gracie, Blending in the extended finite element method by discontinuous Galerkin and assumed strain methods, International Journal for Numerical Methods in Engineering 74 pp 1645– (2008) · Zbl 1195.74175 |

[36] | TarancÃ{\({}^3\)}n, Enhanced blending elements for XFEM applied to linear elastic fracture mechanics, International Journal for Numerical Methods in Engineering 77 pp 126– (2009) |

[37] | Zilian, The enriched spaceâtime finite element method (EST) for simultaneous solution of fluidâstructure interaction, International Journal for Numerical Methods in Engineering 75 pp 305– (2008) · Zbl 1195.74212 |

[38] | MoÃ“s, A computational approach to handle complex microstructure geometries, Computer Methods in Applied Mechanics and Engineering 192 pp 3163– (2003) · Zbl 1054.74056 |

[39] | Sukumar, Modeling holes and inclusions by level sets in the extended finite-element method, Computer Methods in Applied Mechanics and Engineering 190 pp 6183– (2001) · Zbl 1029.74049 |

[40] | Stolarska, Modelling crack growth by level sets in the extended finite element method, International Journal for Numerical Methods in Engineering 51 pp 943– (2001) · Zbl 1022.74049 |

[41] | Hughes, The Finite Element Method: Linear Static and Dynamic Finite Element Analysis (1987) · Zbl 0634.73056 |

[42] | Belytschko, Nonlinear Finite Elements for Continua and Structures (2000) |

[43] | Mariani, Extended finite element method for quasi-brittle fracture, International Journal for Numerical Methods in Engineering 58 pp 103– (2003) · Zbl 1032.74673 |

[44] | Duarte, A high-order generalized FEM for through-the-thickness branched cracks, International Journal for Numerical Methods in Engineering 72 pp 325– (2007) · Zbl 1194.74385 |

[45] | Daux, Arbitrary branched and intersecting cracks with the extended finite element method, International Journal for Numerical Methods in Engineering 48 pp 1741– (2000) · Zbl 0989.74066 |

[46] | Fries, On time integration in the XFEM, International Journal for Numerical Methods in Engineering (2009) · Zbl 1171.76418 |

[47] | Liu, Meshless Methods (2002) |

[48] | Donea, Finite Element Methods for Flow Problems (2003) |

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