High‐order extended finite element method for cracked domains

The aim of the paper is to study the capabilities of the Extended Finite Element Method (XFEM) to achieve accurate computations in non smooth situations such as crack problems. Although the XFEM method ensures a weaker error than classical finite element methods, the rate of convergence is not improved when the mesh parameter h is going to zero because of the presence of a singularity. The difficulty can be overcome by modifying the enrichment of the finite element basis with the asymptotic crack tip displacement solutions as well as with the Heaviside function. Numerical simulations show that the modified XFEM method achieves an optimal rate of convergence (i.e. like in a standard finite element method for a smooth problem)

[1]  I. Babuska,et al.  The generalized finite element method , 2001 .

[2]  I. Babuska,et al.  The Partition of Unity Method , 1997 .

[3]  I. Babuska,et al.  Special finite element methods for a class of second order elliptic problems with rough coefficients , 1994 .

[4]  Ted Belytschko,et al.  Elastic crack growth in finite elements with minimal remeshing , 1999 .

[5]  N. Moës,et al.  Improved implementation and robustness study of the X‐FEM for stress analysis around cracks , 2005 .

[6]  Surendra P. Shah,et al.  Fracture Mechanics of Concrete: Applications of Fracture Mechanics to Concrete, Rock and Other Quasi-Brittle Materials , 1995 .

[7]  I. Babuska,et al.  The design and analysis of the Generalized Finite Element Method , 2000 .

[8]  I. Babuska,et al.  Acta Numerica 2003: Survey of meshless and generalized finite element methods: A unified approach , 2003 .

[9]  T. Belytschko,et al.  On the construction of blending elements for local partition of unity enriched finite elements , 2003 .

[10]  Ted Belytschko,et al.  An extended finite element method for modeling crack growth with frictional contact , 2001 .

[11]  J. Chaboche,et al.  Mechanics of Solid Materials , 1990 .

[12]  Shuodao Wang,et al.  A Mixed-Mode Crack Analysis of Isotropic Solids Using Conservation Laws of Elasticity , 1980 .

[13]  I. Babuska,et al.  The partition of unity finite element method: Basic theory and applications , 1996 .

[14]  P. Grisvard.  Elliptic Problems in Nonsmooth Domains , 1985 .

[15]  J. Prévost,et al.  Modeling quasi-static crack growth with the extended finite element method Part I: Computer implementation , 2003 .

[16]  P. Grisvard.  Singularities in Boundary Value Problems , 1992 .

[17]  Ivo Babuška,et al.  Generalized finite element methods for three-dimensional structural mechanics problems , 2000 .

[18]  T. Belytschko,et al.  Extended finite element method for cohesive crack growth , 2002 .

[19]  C. Sun,et al.  Mechanics of Aircraft Structures , 1998 .

[20]  T. Belytschko,et al.  MODELING HOLES AND INCLUSIONS BY LEVEL SETS IN THE EXTENDED FINITE-ELEMENT METHOD , 2001 .

[21]  Ted Belytschko,et al.  An extended finite element method with higher-order elements for curved cracks , 2003 .

[22]  Ph. Destuynder,et al.  Sur une interpretation math'ematique de l''int'egrale de Rice en th'eorie de la rupture fragile , 1981 .

[23]  Ted Belytschko,et al.  Arbitrary discontinuities in finite elements , 2001 .

[24]  T. Belytschko,et al.  Extended finite element method for three-dimensional crack modelling , 2000 .

[25]  T. Belytschko,et al.  Non‐planar 3D crack growth by the extended finite element and level sets—Part I: Mechanical model , 2002 .

[26]  Ted Belytschko,et al.  A finite element method for crack growth without remeshing , 1999 .

[27]  T. Belytschko,et al.  Non‐planar 3D crack growth by the extended finite element and level sets—Part II: Level set update , 2002 .