An XFEM method for modeling geometrically elaborate crack propagation in brittle materials

We present a method for simulating quasistatic crack propagation in 2-D which combines the extended finite element method (XFEM) with a general algorithm for cutting triangulated domains, and introduce a simple yet general and flexible quadrature rule based on the same geometric algorithm. The combination of these methods gives several advantages. First, the cutting algorithm provides a flexible and systematic way of determining material connectivity, which is required by the XFEM enrichment functions. Also, our integration scheme is straightforward to implement and accurate, without requiring a triangulation that incorporates the new crack edges or the addition of new degrees of freedom to the system. The use of this cutting algorithm and integration rule allows for geometrically complicated domains and complex crack patterns. Copyright © 2011 John Wiley & Sons, Ltd.

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

[2]  T. Belytschko,et al.  A method for multiple crack growth in brittle materials without remeshing , 2004 .

[3]  P. Hansbo,et al.  A finite element method for the simulation of strong and weak discontinuities in solid mechanics , 2004 .

[4]  N. Sukumar,et al.  Generalized Gaussian Quadrature Rules for Discontinuities and Crack Singularities in the Extended Finite Element Method , 2010 .

[5]  D. Chopp,et al.  Three‐dimensional non‐planar crack growth by a coupled extended finite element and fast marching method , 2008 .

[6]  Ted Belytschko,et al.  Cracking node method for dynamic fracture with finite elements , 2009 .

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

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

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

[10]  G. Ventura,et al.  A regularized XFEM model for the transition from continuous to discontinuous displacements , 2008 .

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

[12]  T. Belytschko,et al.  New crack‐tip elements for XFEM and applications to cohesive cracks , 2003 .

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

[14]  Ted Belytschko,et al.  Element‐local level set method for three‐dimensional dynamic crack growth , 2009 .

[15]  Ronald Fedkiw,et al.  A Crystalline, Red Green Strategy for Meshing Highly Deformable Objects with Tetrahedra , 2003, IMR.

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

[17]  Ted Belytschko,et al.  The extended finite element method for fracture in composite materials , 2009 .

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

[19]  Patrick Laborde,et al.  Crack tip enrichment in the XFEM using a cutoff function , 2008 .

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

[21]  T. Belytschko,et al.  Arbitrary branched and intersecting cracks with the eXtended Finite Element Method , 2000 .

[22]  S. Osher,et al.  Algorithms Based on Hamilton-Jacobi Formulations , 1988 .

[23]  J. Melenk.  The Partition of Unity MethodI , 1996 .

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

[25]  David R. Noble,et al.  Quadrature rules for triangular and tetrahedral elements with generalized functions , 2008 .

[26]  Patrick Laborde,et al.  Numerical Analysis A quasi-optimal convergence result for fracture mechanics with XFEM , 2006 .

[27]  Yazid Abdelaziz,et al.  Review: A survey of the extended finite element , 2008 .

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

[29]  Ronald Fedkiw,et al.  Arbitrary cutting of deformable tetrahedralized objects , 2007, SCA '07.

[30]  Isaac Harari,et al.  An efficient finite element method for embedded interface problems , 2009 .

[31]  Ronald Fedkiw,et al.  Level set methods and dynamic implicit surfaces , 2002, Applied mathematical sciences.

[32]  Ted Belytschko,et al.  Fast integration and weight function blending in the extended finite element method , 2009 .

[33]  J. Sethian,et al.  Fronts propagating with curvature-dependent speed: algorithms based on Hamilton-Jacobi formulations , 1988 .

[34]  David L. Chopp,et al.  A hybrid extended finite element/level set method for modeling phase transformations , 2002 .

[35]  M. Duflot.  A study of the representation of cracks with level sets , 2007 .

[36]  Ronald Fedkiw,et al.  A virtual node algorithm for changing mesh topology during simulation , 2004, SIGGRAPH 2004.

[37]  Garth N. Wells,et al.  Cohesive‐zone models, higher‐order continuum theories and reliability methods for computational failure analysis , 2004 .

[38]  S. Bordas,et al.  A simple error estimator for extended finite elements , 2007 .

[39]  Ronald Fedkiw,et al.  Eurographics/ Acm Siggraph Symposium on Computer Animation (2007) Hybrid Simulation of Deformable Solids , 2022 .

[40]  小山 毅,et al.  拡張有限要素法(XFEM)・一般化有限要素法(GFEM)を用いた材料モデリングのレビュー Ted Belytschko,Robert Gracie and Giulio Ventura:A Review of Extended/Generalized Finite Element Methods for Material Modeling [Modeling and Simulations in Materials Science and Engineering, Vol.17, 043001, June 2009](構造,文献抄録) , 2010 .

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

[42]  René de Borst,et al.  Mesh-independent discrete numerical representations of cohesive-zone models , 2006 .

[43]  Jesper L. Asferg,et al.  A consistent partly cracked XFEM element for cohesive crack growth , 2007 .

[44]  Ted Belytschko,et al.  A comment on the article ``A finite element method for simulation of strong and weak discontinuities in solid mechanics'' by A. Hansbo and P. Hansbo [Comput. Methods Appl. Mech. Engrg. 193 (2004) 3523-3540] , 2006 .

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

[46]  Stefano Mariani,et al.  Extended finite element method for quasi‐brittle fracture , 2003 .

[47]  Benoit Prabel,et al.  Level set X‐FEM non‐matching meshes: application to dynamic crack propagation in elastic–plastic media , 2007 .

[48]  Ted Belytschko,et al.  SINGULAR ENRICHMENT FINITE ELEMENT METHOD FOR ELASTODYNAMIC CRACK PROPAGATION , 2004 .

[49]  T. Belytschko,et al.  A method for growing multiple cracks without remeshing and its application to fatigue crack growth , 2004 .

[50]  Bhushan Lal Karihaloo,et al.  Modelling of stationary and growing cracks in FE framework without remeshing: a state-of-the art review , 2003 .

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

[52]  Gary R. Consolazio,et al.  Finite Elements , 2007, Handbook of Dynamic System Modeling.

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

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

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

[56]  G. Ventura.  On the elimination of quadrature subcells for discontinuous functions in the eXtended Finite‐Element Method , 2006 .

[57]  Ted Belytschko,et al.  A method for dynamic crack and shear band propagation with phantom nodes , 2006 .

[58]  Ted Belytschko,et al.  Dynamic Fracture of Shells Subjected to Impulsive Loads , 2009 .