The extended/generalized finite element method: An overview of the method and its applications

An overview of the extended/generalized finite element method (GEFM/XFEM) with emphasis on methodological issues is presented. This method enables the accurate approximation of solutions that involve jumps, kinks, singularities, and other locally non‐smooth features within elements. This is achieved by enriching the polynomial approximation space of the classical finite element method. The GEFM/XFEM has shown its potential in a variety of applications that involve non‐smooth solutions near interfaces: Among them are the simulation of cracks, shear bands, dislocations, solidification, and multi‐field problems. Copyright © 2010 John Wiley & Sons, Ltd.

[1]  D. S. Dugdale.  Yielding of steel sheets containing slits , 1960 .

[2]  M. Williams.  The Bending Stress Distribution at the Base of a Stationary Crack , 1961 .

[3]  G. I. Barenblatt.  THE MATHEMATICAL THEORY OF EQUILIBRIUM CRACKS IN BRITTLE FRACTURE , 1962 .

[4]  J. T. Oden,et al.  On the calculation of consistent stress distributions in finite element approximations , 1971 .

[5]  C. D. Mote.  Global‐local finite element , 1971 .

[6]  I. Babuska.  Error-bounds for finite element method , 1971 .

[7]  F. Brezzi.  On the existence, uniqueness and approximation of saddle-point problems arising from lagrangian multipliers , 1974 .

[8]  R. Barsoum.  Application of quadratic isoparametric finite elements in linear fracture mechanics , 1974 .

[9]  Milton E. Rose,et al.  Weak-element approximations to elliptic differential equations , 1975 .

[10]  A. Hillerborg,et al.  Analysis of crack formation and crack growth in concrete by means of fracture mechanics and finite elements , 1976 .

[11]  J. Z. Zhu,et al.  The finite element method , 1977 .

[12]  P. Lancaster,et al.  Surfaces generated by moving least squares methods , 1981 .

[13]  I. Babuska,et al.  Generalized Finite Element Methods: Their Performance and Their Relation to Mixed Methods , 1983 .

[14]  Charles I. Goldstein,et al.  The weak element method applied to Helmholtz type equations , 1986 .

[15]  Ahmed K. Noor,et al.  Global-local methodologies and their application to nonlinear analysis , 1986 .

[16]  Gen-Hua Shi,et al.  Manifold Method of Material Analysis , 1992 .

[17]  J. Z. Zhu,et al.  The superconvergent patch recovery and a posteriori error estimates. Part 2: Error estimates and adaptivity , 1992 .

[18]  J. Z. Zhu,et al.  The superconvergent patch recovery and a posteriori error estimates. Part 1: The recovery technique , 1992 .

[19]  T. Belytschko,et al.  Finite element derivative recovery by moving least square interpolants , 1994 .

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

[21]  Ivo Babuška,et al.  A Generalized Finite Element Method for solving the Helmholtz equation in two dimensions with minimal pollution , 1995 .

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

[23]  Mark A Fleming,et al.  Meshless methods: An overview and recent developments , 1996 .

[24]  Oden,et al.  An h-p adaptive method using clouds , 1996 .

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

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

[27]  J A Sethian,et al.  A fast marching level set method for monotonically advancing fronts. , 1996, Proceedings of the National Academy of Sciences of the United States of America.

[28]  Mark A Fleming,et al.  ENRICHED ELEMENT-FREE GALERKIN METHODS FOR CRACK TIP FIELDS , 1997 .

[29]  O. Cessenat,et al.  Application of an Ultra Weak Variational Formulation of Elliptic PDEs to the Two-Dimensional Helmholtz Problem , 1998 .

[30]  Ted Belytschko,et al.  EFG approximation with discontinuous derivatives , 1998 .

[31]  Guangqi Chen,et al.  DEVELOPMENT OF HIGH-ORDER MANIFOLD METHOD , 1998 .

[32]  S. Osher,et al.  A PDE-Based Fast Local Level Set Method 1 , 1998 .

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

[34]  Peter Monk,et al.  A least-squares method for the Helmholtz equation , 1999 .

[35]  Pierre Ladevèze,et al.  Nonlinear Computational Structural Mechanics , 1999 .

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

[37]  J. A. Sethian,et al.  Fast Marching Methods , 1999, SIAM Rev..

[38]  S. Osher,et al.  Regular Article: A PDE-Based Fast Local Level Set Method , 1999 .

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

[40]  James A. Sethian,et al.  Level Set Methods and Fast Marching Methods , 1999 .

[41]  Mark Sussman,et al.  An Efficient, Interface-Preserving Level Set Redistancing Algorithm and Its Application to Interfacial Incompressible Fluid Flow , 1999, SIAM J. Sci. Comput..

[42]  J. Oden,et al.  A Posteriori Error Estimation in Finite Element Analysis , 2000 .

[43]  J. Oden,et al.  A Posteriori Error Estimation in Finite Element Analysis: Oden/A Posteriori , 2000 .

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

[45]  P. Smereka,et al.  A Remark on Computing Distance Functions , 2000 .

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

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

[48]  Ted Belytschko,et al.  Modeling fracture in Mindlin–Reissner plates with the extended finite element method , 2000 .

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

[50]  T. Strouboulis,et al.  The generalized finite element method: an example of its implementation and illustration of its performance , 2000 .

[51]  C. Farhat,et al.  The Discontinuous Enrichment Method , 2000 .

[52]  Ted Belytschko,et al.  Discontinuous enrichment in finite elements with a partition of unity method , 2000 .

[53]  Wing Kam Liu,et al.  Nonlinear Finite Elements for Continua and Structures , 2000 .

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

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

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

[57]  Ted Belytschko,et al.  Modelling crack growth by level sets in the extended finite element method , 2001 .

[58]  S. Osher,et al.  Level set methods: an overview and some recent results , 2001 .

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

[60]  Ted Belytschko,et al.  The extended finite element method for rigid particles in Stokes flow , 2001 .

[61]  Charbel Farhat,et al.  Partitioned analysis of coupled mechanical systems , 2001 .

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

[63]  L. J. Sluys,et al.  A new method for modelling cohesive cracks using finite elements , 2001 .

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

[65]  P. Hansbo,et al.  An unfitted finite element method, based on Nitsche's method, for elliptic interface problems , 2002 .

[66]  John E. Dolbow,et al.  Solving thermal and phase change problems with the eXtended finite element method , 2002 .

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

[68]  Michael Griebel,et al.  A Particle-Partition of Unity Method-Part II: Efficient Cover Construction and Reliable Integration , 2001, SIAM J. Sci. Comput..

[69]  R. de Borst,et al.  Simulating the propagation of displacement discontinuities in a regularized strain‐softening medium , 2002 .

[70]  Ted Belytschko,et al.  A vector level set method and new discontinuity approximations for crack growth by EFG , 2002 .

[71]  Ted Belytschko,et al.  Structured extended finite element methods for solids defined by implicit surfaces , 2002 .

[72]  T. Belytschko,et al.  The extended finite element method (XFEM) for solidification problems , 2002 .

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

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

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

[76]  T. Belytschko,et al.  Topology optimization with implicit functions and regularization , 2003 .

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

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

[79]  Charbel Farhat,et al.  The discontinuous enrichment method for multiscale analysis , 2003 .

[80]  M. Jirásek,et al.  Process zone resolution by extended finite elements , 2003 .

[81]  A. Needleman,et al.  A cohesive segments method for the simulation of crack growth , 2003 .

[82]  Michael Griebel,et al.  A Particle-Partition of Unity Method Part V: Boundary Conditions , 2003 .

[83]  D. Chopp,et al.  Fatigue crack propagation of multiple coplanar cracks with the coupled extended finite element/fast marching method , 2003 .

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

[85]  Lothar Gaul,et al.  Boundary element methods for engineers and scientists , 2003 .

[86]  T. Belytschko,et al.  Dynamic crack propagation based on loss of hyperbolicity and a new discontinuous enrichment , 2003 .

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

[88]  Jean-François Remacle,et al.  A computational approach to handle complex microstructure geometries , 2003 .

[89]  D. Chopp,et al.  Extended finite element method and fast marching method for three-dimensional fatigue crack propagation , 2003 .

[90]  Endel V. Iarve,et al.  Mesh independent modelling of cracks by using higher order shape functions , 2003 .

[91]  Li Ge-wen.  Fast Tree-Based Redistancing for Level Set Computations , 2003 .

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

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

[94]  A. Huerta,et al.  Finite Element Methods for Flow Problems , 2003 .

[95]  T. Belytschko,et al.  An Extended Finite Element Method for Two-Phase Fluids , 2003 .

[96]  I. Babuska,et al.  Generalized finite element method using mesh-based handbooks: application to problems in domains with many voids , 2003 .

[97]  David L. Chopp,et al.  Modeling thermal fatigue cracking in integrated circuits by level sets and the extended finite element method , 2003 .

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

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

[100]  T. Belytschko,et al.  An enriched finite element method and level sets for axisymmetric two‐phase flow with surface tension , 2003 .

[101]  Bhushan Lal Karihaloo,et al.  Direct evaluation of accurate coefficients of the linear elastic crack tip asymptotic field , 2003 .

[102]  T. Belytschko,et al.  Vector level sets for description of propagating cracks in finite elements , 2003 .

[103]  Charbel Farhat,et al.  A discontinuous Galerkin method with Lagrange multipliers for the solution of Helmholtz problems in the mid-frequency regime , 2003 .

[104]  Paul Steinmann,et al.  A hybrid discontinuous Galerkin/interface method for the computational modelling of failure , 2004 .

[105]  Charbel Farhat,et al.  Higher‐order extensions of a discontinuous Galerkin method for mid‐frequency Helmholtz problems , 2004 .

[106]  Simon K. Warfield,et al.  On Extended Finite Element Method (XFEM) for Modelling of Organ Deformations Associated with Surgical Cuts , 2004, ISMS.

[107]  S. Rahman,et al.  An enriched meshless method for non‐linear fracture mechanics , 2004 .

[108]  C. S. de Barcellos,et al.  On error estimator and p‐adaptivity in the generalized finite element method , 2004 .

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

[110]  John E. Dolbow,et al.  On strategies for enforcing interfacial constraints and evaluating jump conditions with the extended finite element method , 2004 .

[111]  H. Matthies,et al.  Classification and Overview of Meshfree Methods , 2004 .

[112]  Antonio Huerta,et al.  Imposing essential boundary conditions in mesh-free methods , 2004 .

[113]  Ivo Babuška,et al.  p‐version of the generalized FEM using mesh‐based handbooks with applications to multiscale problems , 2004 .

[114]  Bhushan Lal Karihaloo,et al.  XFEM for direct evaluation of mixed mode SIFs in homogeneous and bi‐materials , 2004 .

[115]  Ted Belytschko,et al.  Arbitrary discontinuities in space–time finite elements by level sets and X‐FEM , 2004 .

[116]  C. S. de Barcellos,et al.  Generalized finite element method in structural nonlinear analysis – a p-adaptive strategy , 2004 .

[117]  I. Babuska,et al.  GENERALIZED FINITE ELEMENT METHODS — MAIN IDEAS, RESULTS AND PERSPECTIVE , 2004 .

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

[119]  Ted Belytschko,et al.  Dislocations by partition of unity , 2005 .

[120]  Julien Réthoré,et al.  An energy‐conserving scheme for dynamic crack growth using the eXtended finite element method , 2005 .

[121]  A. Huerta,et al.  Finite Element Methods for Flow Problems , 2003 .

[122]  T. Belytschko,et al.  Continuum–discontinuum modelling of shear bands , 2005 .

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

[124]  P. Wesseling,et al.  A mass‐conserving Level‐Set method for modelling of multi‐phase flows , 2005 .

[125]  Julien Réthoré,et al.  A combined space–time extended finite element method , 2005 .

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

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

[128]  T. Belytschko,et al.  Strong and weak arbitrary discontinuities in spectral finite elements , 2005 .

[129]  Alain Combescure,et al.  Appropriate extended functions for X-FEM simulation of plastic fracture mechanics , 2006 .

[130]  T. Belytschko,et al.  Two‐scale shear band evolution by local partition of unity , 2006 .

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

[132]  Ted Belytschko,et al.  A local space-time discontinuous finite element method , 2006 .

[133]  C. Farhat,et al.  The discontinuous enrichment method for elastic wave propagation in the medium‐frequency regime , 2006 .

[134]  Ivo Babuška,et al.  A posteriori error estimation for generalized finite element methods , 2006 .

[135]  Ivo Babuška,et al.  The generalized finite element method for Helmholtz equation: Theory, computation, and open problems , 2006 .

[136]  Charbel Farhat,et al.  Three‐dimensional discontinuous Galerkin elements with plane waves and Lagrange multipliers for the solution of mid‐frequency Helmholtz problems , 2006 .

[137]  T. Belytschko,et al.  An Eulerian–Lagrangian method for fluid–structure interaction based on level sets , 2006 .

[138]  Tae-Yeon Kim,et al.  A mortared finite element method for frictional contact on arbitrary interfaces , 2006 .

[139]  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 .

[140]  Carlos Armando Duarte,et al.  A Generalized Finite Element Method for polycrystals with discontinuous grain boundaries , 2006 .

[141]  Bhushan Lal Karihaloo,et al.  Improving the accuracy of XFEM crack tip fields using higher order quadrature and statically admissible stress recovery , 2006 .

[142]  Julien Réthoré,et al.  Efficient explicit time stepping for the eXtended Finite Element Method (X‐FEM) , 2006 .

[143]  T. Belytschko,et al.  The intrinsic XFEM: a method for arbitrary discontinuities without additional unknowns , 2006 .

[144]  B. Karihaloo,et al.  Incremental‐secant modulus iteration scheme and stress recovery for simulating cracking process in quasi‐brittle materials using XFEM , 2007 .

[145]  Marc Duflot,et al.  Derivative recovery and a posteriori error estimate for extended finite elements , 2007 .

[146]  Alireza Asadpoure,et al.  Developing new enrichment functions for crack simulation in orthotropic media by the extended finite element method , 2007 .

[147]  Stéphane Roux,et al.  Shear-band capturing using a multiscale extended digital image correlation technique , 2007 .

[148]  Ted Belytschko,et al.  The intrinsic partition of unity method , 2007 .

[149]  T. Belytschko,et al.  Two‐scale method for shear bands: thermal effects and variable bandwidth , 2007 .

[150]  G. Meschke,et al.  Energy-based modeling of cohesive and cohesionless cracks via X-FEM , 2007 .

[151]  Carlos Armando Duarte,et al.  A high‐order generalized FEM for through‐the‐thickness branched cracks , 2007 .

[152]  Isaac Harari,et al.  A bubble‐stabilized finite element method for Dirichlet constraints on embedded interfaces , 2007 .

[153]  Ted Belytschko,et al.  On XFEM applications to dislocations and interfaces , 2007 .

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

[155]  Arnold Reusken,et al.  An extended pressure finite element space for two-phase incompressible flows with surface tension , 2007, J. Comput. Phys..

[156]  S. Eckardt,et al.  Modelling of cohesive crack growth in concrete structures with the extended finite element method , 2007 .

[157]  C. Comi,et al.  Extended finite element simulation of quasi-brittle fracture in functionally graded materials , 2007 .

[158]  Stéphane Bordas,et al.  An extended finite element library , 2007 .

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

[160]  Ted Belytschko,et al.  A new fast finite element method for dislocations based on interior discontinuities , 2007 .

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

[162]  A. Legay,et al.  An enriched finite element approach for prescribed motions of thin immersed structures in a fluid , 2007 .

[163]  David L. Chopp,et al.  The extended finite element method for boundary layer problems in biofilm growth , 2007 .

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

[165]  Wolfgang A. Wall,et al.  Enhancement of fixed‐grid methods towards complex fluid–structure interaction applications , 2008 .

[166]  John E. Dolbow,et al.  Residual-free bubbles for embedded Dirichlet problems , 2008 .

[167]  T. Belytschko,et al.  Blending in the extended finite element method by discontinuous Galerkin and assumed strain methods , 2008 .

[168]  Soheil Mohammadi,et al.  Extended Finite Element Method: for Fracture Analysis of Structures , 2008 .

[169]  Nicolas Moës,et al.  Mass lumping strategies for X‐FEM explicit dynamics: Application to crack propagation , 2008 .

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

[171]  D. Chopp,et al.  A combined extended finite element and level set method for biofilm growth , 2008 .

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

[173]  Ronaldo I. Borja,et al.  A contact algorithm for frictional crack propagation with the extended finite element method , 2008 .

[174]  W. Wall,et al.  An eXtended Finite Element Method/Lagrange multiplier based approach for fluid-structure interaction , 2008 .

[175]  Andreas Zilian,et al.  The enriched space–time finite element method (EST) for simultaneous solution of fluid–structure interaction , 2008 .

[176]  Alok Sutradhar,et al.  Symmetric Galerkin Boundary Element Method , 2008 .

[177]  C. Farhat,et al.  A discontinuous enrichment method for three‐dimensional multiscale harmonic wave propagation problems in multi‐fluid and fluid–solid media , 2008 .

[178]  Ted Belytschko,et al.  Parametric enrichment adaptivity by the extended finite element method , 2008 .

[179]  Ionel Nistor,et al.  Numerical implementation of the eXtended Finite Element Method for dynamic crack analysis , 2008, Adv. Eng. Softw..

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

[181]  Elena Benvenuti,et al.  A regularized XFEM framework for embedded cohesive interfaces , 2008 .

[182]  Ted Belytschko,et al.  On a new extended finite element method for dislocations: Core enrichment and nonlinear formulation , 2008 .

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

[184]  Ivo Babuška,et al.  The generalized finite element method for Helmholtz equation. Part II: Effect of choice of handbook functions, error due to absorbing boundary conditions and its assessment , 2008 .

[185]  Juan José Ródenas,et al.  A recovery‐type error estimator for the extended finite element method based on singular+smooth stress field splitting , 2008 .

[186]  S. Bordas,et al.  A posteriori error estimation for extended finite elements by an extended global recovery , 2008 .

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

[188]  Ted Belytschko,et al.  An extended finite element method for dislocations in complex geometries: Thin films and nanotubes , 2009 .

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

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

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

[192]  Thomas-Peter Fries,et al.  Higher‐order XFEM for curved strong and weak discontinuities , 2009 .

[193]  Eugenio Giner,et al.  An Abaqus implementation of the extended finite element method , 2009 .

[194]  Thomas-Peter Fries,et al.  A localized mixed‐hybrid method for imposing interfacial constraints in the extended finite element method (XFEM) , 2009 .

[195]  T. Fries,et al.  On time integration in the XFEM , 2009 .

[196]  Ted Belytschko,et al.  Concurrently coupled atomistic and XFEM models for dislocations and cracks , 2009 .

[197]  J. Cox.  An extended finite element method with analytical enrichment for cohesive crack modeling , 2009 .

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

[199]  Thomas-Peter Fries,et al.  The intrinsic XFEM for two‐fluid flows , 2009 .

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

[201]  Lenka Jerábková,et al.  Stable Cutting of Deformable Objects in Virtual Environments Using XFEM , 2009, IEEE Computer Graphics and Applications.

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

[203]  Stéphane Bordas,et al.  Numerical integration over arbitrary polygonal domains based on Schwarz–Christoffel conformal mapping , 2009 .

[204]  Charbel Farhat,et al.  A discontinuous enrichment method for the finite element solution of high Péclet advection-diffusion problems , 2009 .

[205]  T. Fries,et al.  The XFEM for high‐gradient solutions in convection‐dominated problems , 2009 .

[206]  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 .

[207]  T. Rabczuk,et al.  On three-dimensional modelling of crack growth using partition of unity methods , 2010 .

[208]  Jean-François Remacle,et al.  Imposing Dirichlet boundary conditions in the eXtended Finite Element Method , 2011 .