Strain smoothing in FEM and XFEM

We present in this paper recent achievements realised on the application of strain smoothing in finite elements and propose suitable extensions to problems with discontinuities and singularities. The numerical results indicate that for 2D and 3D continuum, locking can be avoided. New plate and shell formulations that avoid both shear and membrane locking are also briefly reviewed. The principle is then extended to partition of unity enrichment to simplify numerical integration of discontinuous approximations in the extended finite element method. Examples are presented to test the new elements for problems involving cracks in linear elastic continua and cracked plates. In the latter case, the proposed formulation suppresses locking and yields elements which behave very well, even in the thin plate limit. Two important features of the set of elements presented are their insensitivity to mesh distortion and a lower computational cost than standard finite elements for the same accuracy. These elements are easily implemented in existing codes since they only require the modification of the discretized gradient operator, B.

[1]  Jiun-Shyan Chen,et al.  A stabilized conforming nodal integration for Galerkin mesh-free methods , 2001 .

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

[3]  T. Rabczuk,et al.  A meshfree thin shell method for non‐linear dynamic fracture , 2007 .

[4]  Eduardo N. Dvorkin,et al.  A formulation of general shell elements—the use of mixed interpolation of tensorial components† , 1986 .

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

[6]  H. Saunders.  Book Reviews : NUMERICAL METHODS IN FINITE ELEMENT ANALYSIS K.-J. Bathe and E.L. Wilson Prentice-Hall, Inc, Englewood Cliffs, NJ , 1978 .

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

[8]  Michael Ortiz,et al.  Local Maximum-Entropy Approximation Schemes , 2007 .

[9]  E. Stein,et al.  On the duality of finite element discretization error control in computational Newtonian and Eshelbian mechanics , 2007 .

[10]  K. Bathe,et al.  A four‐node plate bending element based on Mindlin/Reissner plate theory and a mixed interpolation , 1985 .

[11]  T. Belytschko,et al.  Physical stabilization of the 4-node shell element with one point quadrature , 1994 .

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

[13]  H. T. Corten,et al.  A mixed-mode crack analysis of rectilinear anisotropic solids using conservation laws of elasticity , 1980 .

[14]  Thomas Zimmermann,et al.  Object-oriented nonlinear finite element programming: a primer , 2000 .

[15]  Marc Alexander Schweitzer,et al.  Partition of Unity Method , 2003 .

[16]  Marc Duflot,et al.  Meshless methods: A review and computer implementation aspects , 2008, Math. Comput. Simul..

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

[18]  K. Y. Dai,et al.  Theoretical aspects of the smoothed finite element method (SFEM) , 2007 .

[19]  Jean-François Remacle,et al.  A substructured FE‐shell/XFE‐3D method for crack analysis in thin‐walled structures , 2007 .

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

[21]  T. Rabczuk,et al.  A Meshfree Thin Shell for Arbitrary Evolving Cracks Based on An Extrinsic Basis , 2006 .

[22]  T. Belytschko,et al.  Analysis of thin shells by the Element-Free Galerkin method , 1996 .

[23]  Stéphane Bordas,et al.  Enriched finite elements and level sets for damage tolerance assessment of complex structures , 2006 .

[24]  H. Nguyen-Xuan,et al.  A geometrically non-linear three-dimensional cohesive crack method for reinforced concrete structures , 2008 .

[25]  D. Malkus,et al.  Mixed finite element methods—reduced and selective integration techniques: a unification of concepts , 1990 .

[26]  N. Sukumar,et al.  Conforming polygonal finite elements , 2004 .

[27]  H. Askes,et al.  Error estimation and p -adaptivity based on the partition of unity method , 2002 .

[28]  G. W. Rankin,et al.  Computational study of flow past a cylinder with combined in-line and transverse oscillation , 1995 .

[29]  H. Nguyen-Xuan,et al.  A smoothed finite element method for plate analysis , 2008 .

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

[31]  Thomas J. R. Hughes,et al.  Nonlinear finite element analysis of shells: Part I. three-dimensional shells , 1981 .

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

[33]  Brian Moran,et al.  Crack tip and associated domain integrals from momentum and energy balance , 1987 .

[34]  K. Y. Lam,et al.  Selective smoothed finite element method , 2007 .

[35]  Jean-François Remacle,et al.  Substructuring FE-XFE approaches applied to three-dimensional crack propagation , 2008 .

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

[37]  Jean-François Remacle,et al.  A substructured FE/XFE method for stress intensity factors computation in an industrial structure , 2007 .

[38]  H. M. Boduroglu,et al.  Internal and edge cracks in a plate of finite width under bending , 1983 .

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

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

[41]  J. Dolbow,et al.  Enrichment of enhanced assumed strain approximations for representing strong discontinuities: addressing volumetric incompressibility and the discontinuous patch test , 2004 .

[42]  Stéphane Bordas,et al.  Smooth finite element methods: Convergence, accuracy and properties , 2008 .

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

[44]  N. Sukumar,et al.  Maximum Entropy Approximation , 2005 .

[45]  Satya N. Atluri,et al.  A locking-free meshless local Petrov-Galerkin formulation for thick and thin plates , 2005 .

[46]  J. Rice.  A path-independent integral and the approximate analysis of strain , 1968 .

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

[48]  David W. Murray,et al.  Nonlinear Finite Element Analysis of Steel Frames , 1983 .

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

[50]  B. Moran,et al.  Stabilized conforming nodal integration in the natural‐element method , 2004 .

[51]  N. Sukumar,et al.  Extended finite element method on polygonal and quadtree meshes , 2008 .

[52]  Miloš Zlámal,et al.  Superconvergence and reduced integration in the finite element method , 1978 .

[53]  Robert L. Taylor,et al.  Resultant fields for mixed plate bending elements , 1990 .

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

[55]  C. Duarte,et al.  hp-Clouds in Mindlin's thick plate model , 2000 .

[56]  Eric Wyart,et al.  Three-dimensional crack analysis in aeronautical structures using the substructured finite element / extended finite element method , 2007 .

[57]  Magdalena Ortiz,et al.  Local maximum‐entropy approximation schemes: a seamless bridge between finite elements and meshfree methods , 2006 .

[58]  T. Rabczuk,et al.  Three-dimensional crack initiation, propagation, branching and junction in non-linear materials by an extended meshfree method without asymptotic enrichment , 2008 .

[59]  Eric P. Kasper,et al.  A mixed-enhanced strain method , 2000 .

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

[61]  T. Belytschko,et al.  Element‐free Galerkin methods , 1994 .

[62]  Jean-Louis Batoz,et al.  Evaluation of a new quadrilateral thin plate bending element , 1982 .

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

[64]  E. Stein,et al.  Adaptive finite element analysis of crack propagation in elastic fracture mechanics based on averaging techniques , 2004 .

[65]  R. Hill.  The mathematical theory of plasticity , 1950 .

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

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

[68]  Stéphane Bordas,et al.  New trends in the development of the X-FEM, chapter architecture trade-offs including a mesher in an object-oriented extended finite element code , 2007 .

[69]  Ted Belytschko,et al.  Analysis of thin plates by the element-free Galerkin method , 1995 .

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

[71]  J. P. Moitinho de Almeida,et al.  A SET OF HYBRID EQUILIBRIUM FINITE ELEMENT MODELS FOR THE ANALYSIS OF THREE-DIMENSIONAL SOLIDS , 1996 .

[72]  D. Owen,et al.  Finite elements in plasticity : theory and practice , 1980 .

[73]  Robert D. Cook,et al.  Improved Two-Dimensional Finite Element , 1974 .

[74]  J. C. Simo,et al.  On a stress resultant geometrically exact shell model. Part III: computational aspects of the nonlinear theory , 1990 .

[75]  T. Belytschko,et al.  THE NATURAL ELEMENT METHOD IN SOLID MECHANICS , 1998 .

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

[77]  N. Sukumar.  Construction of polygonal interpolants: a maximum entropy approach , 2004 .

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

[79]  K. Bathe,et al.  A continuum mechanics based four‐node shell element for general non‐linear analysis , 1984 .

[80]  Sergey Korotov,et al.  Goal-Oriented Error Estimates Based on Different FE-Spaces for the Primal and the Dual Problem with Applications to Fracture Mechanics , 2007 .

[81]  M. A. Crisfield,et al.  Finite elements in plasticity—theory and practice, D. R. J. Owen and E. Hinton, Pineridge Press, Swansea , 1981 .

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

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

[84]  Grégory Legrain,et al.  Stability of incompressible formulations enriched with X-FEM , 2008 .

[85]  K. Bathe.  Finite Element Procedures , 1995 .

[86]  Dongdong Wang,et al.  Locking-free stabilized conforming nodal integration for meshfree Mindlin-Reissner plate formulation , 2004 .

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

[88]  K. Y. Dai,et al.  An n-sided polygonal smoothed finite element method (nSFEM) for solid mechanics , 2007 .

[89]  T. Rabczuk,et al.  A three-dimensional meshfree method for continuous multiple-crack initiation, propagation and junction in statics and dynamics , 2007 .

[90]  J. C. Simo,et al.  On a stress resultant geometrically exact shell model. Part II: the linear theory; computational aspects , 1989 .

[91]  James G. Conley,et al.  A simulation-based design paradigm for complex cast components , 2007, Engineering with Computers.

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

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

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

[95]  Andrzej Truty,et al.  Stabilized finite elements applied to elastoplasticity: I. Mixed displacement–pressure formulation , 2004 .

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

[97]  R. Wright,et al.  Overview and construction of meshfree basis functions: from moving least squares to entropy approximants , 2007 .

[98]  Erwin Stein,et al.  Goal-oriented a posteriori error estimates in linear elastic fracture mechanics , 2006 .

[99]  Satya N. Atluri,et al.  Meshless Local Petrov-Galerkin (MLPG) Formulation for Analysis of Thick Plates , 2004 .

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