On the enrichment zone size for optimal convergence rate of the Generalized/Extended Finite Element Method

Singular enrichment functions are broadly used in Generalized or Extended Finite Element Methods (GFEM/XFEM) for linear elastic fracture mechanics problems. These functions are used at finite element nodes within an enrichment zone around the crack tip/front in 2- and 3-D problems, respectively. Small zones lead to suboptimal convergence rate of the method while large ones lead to ill-conditioning of the system of equations and to a large number of degrees of freedom. This paper presents an a priori estimate for the minimum size of the enrichment zone required for optimal convergence rate of the GFEM/XFEM. The estimate shows that the minimum size of the enrichment zone for optimal convergence rate depends on the element size and polynomial order of the GFEM/XFEM shape functions. Detailed numerical verification of these findings is also presented.

[1]  Varun Gupta,et al.  Improved conditioning and accuracy of GFEM/XFEM for three-dimensional fracture mechanics , 2015 .

[2]  I. Babuska,et al.  Finite Element Analysis , 2021 .

[3]  Eugenio Giner,et al.  Enhanced blending elements for XFEM applied to linear elastic fracture mechanics , 2009 .

[4]  T. Belytschko,et al.  A review of extended/generalized finite element methods for material modeling , 2009 .

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

[6]  Jean-Herve Prevost,et al.  MODELING QUASI-STATIC CRACK GROWTH WITH THE EXTENDED FINITE ELEMENT METHOD PART II: NUMERICAL APPLICATIONS , 2003 .

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

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

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

[10]  Serge Nicaise,et al.  Optimal convergence analysis for the extended finite element method , 2011 .

[11]  J. Tinsley Oden,et al.  An hp Adaptive Method Using Clouds C , 2006 .

[12]  Eitan Grinspun,et al.  Harmonic enrichment functions: A unified treatment of multiple, intersecting and branched cracks in the extended finite element method , 2010 .

[13]  Xiangmin Jiao,et al.  hp‐Generalized FEM and crack surface representation for non‐planar 3‐D cracks , 2009 .

[14]  L. Knöfel,et al.  Oden, J. T. / Reddy, J. N., An Introduction to the Mathematical Theory of Finite Elements. New York‐London‐Sydney‐Toronto. John Wiley & Sons. 1976. XII, 429 S., £ 17.50. $ 32.00 , 1978 .

[15]  E. Grinspun,et al.  Harmonic Enrichment Functions : A Unified Treatment of Multiple, Intersecting, Branched Cracks , 2010 .

[16]  Marc Alexander Schweitzer,et al.  Stable enrichment and local preconditioning in the particle-partition of unity method , 2011, Numerische Mathematik.

[17]  Michael Griebel,et al.  A Particle-Partition of Unity Method for the Solution of Elliptic, Parabolic, and Hyperbolic PDEs , 2000, SIAM J. Sci. Comput..

[18]  Glaucio H. Paulino,et al.  Integration of singular enrichment functions in the generalized/extended finite element method for three‐dimensional problems , 2009 .

[19]  T. Fries.  A corrected XFEM approximation without problems in blending elements , 2008 .

[20]  Michel Salaün,et al.  High‐order extended finite element method for cracked domains , 2005 .

[21]  I. Babuska,et al.  Stable Generalized Finite Element Method (SGFEM) , 2011, 1104.0960.

[22]  T. Belytschko,et al.  Analysis of three‐dimensional crack initiation and propagation using the extended finite element method , 2005 .

[23]  T. Liszka,et al.  A generalized finite element method for the simulation of three-dimensional dynamic crack propagation , 2001 .

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

[25]  O. C. Zienkiewicz,et al.  A new cloud-based hp finite element method , 1998 .

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

[27]  Roland Glowinski,et al.  An introduction to the mathematical theory of finite elements , 1976 .

[28]  Ivo Babuška,et al.  A stable and optimally convergent generalized FEM (SGFEM) for linear elastic fracture mechanics , 2013 .

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

[30]  Carlos Armando Duarte,et al.  Extraction of stress intensity factors from generalized finite element solutions , 2005 .

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

[32]  J. Oden,et al.  H‐p clouds—an h‐p meshless method , 1996 .