Construct ‘FE-Meshfree’ Quad4 using mean value coordinates

Abstract The present work uses mean value coordinates to construct the shape functions of a hybrid ‘FE-Meshfree’ quadrilateral element, which is named as Quad4-MVC. This Quad4-MVC can be regarded as the development of the ‘FE-Meshfree’ quadrilateral element with radial-polynomial point interpolation (Quad4-RPIM). Similar to Quad4-RPIM, Quad4-MVC has Kronecker delta property on the boundaries of computational domain, so essential boundary conditions can be enforced as conveniently as in the finite element method (FEM). The novelty of the present work is to construct nodal approximations using mean value coordinates, instead of radial basis functions which are used in Quad4-RPIM. Compared to the radial basis functions, mean value coordinates does not utilize any uncertain parameters, which enhances stability of numerical results. Numerical tests in this paper show that the performance of Quad4-RPIM becomes even worse than four-node iso-parametric element (Quad4) when the parameters of radial basis functions are not chosen properly. However, the performance of Quad4-MVC is stably better than Quad4.

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

[2]  John Middleton,et al.  Finite element modelling of maxillofacial surgery and facial expressions—a preliminary study , 2010, The international journal of medical robotics + computer assisted surgery : MRCAS.

[3]  Yinghua Liu,et al.  DYNAMIC ELASTOPLASTIC ANALYSIS USING THE MESHLESS LOCAL NATURAL NEIGHBOR INTERPOLATION METHOD , 2011 .

[4]  S. Rajendran,et al.  A partition-of-unity based ‘FE-Meshfree’ QUAD4 element with radial-polynomial basis functions for static analyses , 2011 .

[5]  Robert W. Zimmerman,et al.  Energy conservative property of impulse‐based methods for collision resolution , 2013 .

[6]  R. L. Hardy.  Multiquadric equations of topography and other irregular surfaces , 1971 .

[7]  S. Atluri,et al.  A new Meshless Local Petrov-Galerkin (MLPG) approach in computational mechanics , 1998 .

[8]  T. Belytschko,et al.  Stable particle methods based on Lagrangian kernels , 2004 .

[9]  J. Monaghan.  Smoothed particle hydrodynamics , 2005 .

[10]  Brian Moran,et al.  A computational model for nucleation of solid-solid phase transformations , 1995 .

[11]  张建海,et al.  A novel four-node quadrilateral element with continuous nodal stress , 2009 .

[12]  Guirong Liu,et al.  A face-based smoothed finite element method (FS-FEM) for visco-elastoplastic analyses of 3D solids using tetrahedral mesh , 2009 .

[13]  Shenshen Chen,et al.  A meshless local natural neighbour interpolation method for analysis of two-dimensional piezoelectric structures , 2013 .

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

[15]  S. C. Wu,et al.  The Mean Value Method for Crack Propagation , 2009, 2009 Fifth International Conference on Natural Computation.

[16]  Guirong Liu,et al.  An edge-based smoothed finite element method (ES-FEM) for static, free and forced vibration analyses of solids , 2009 .

[17]  K. Y. Dai,et al.  A Smoothed Finite Element Method for Mechanics Problems , 2007 .

[18]  Ted Belytschko,et al.  Cracking particles: a simplified meshfree method for arbitrary evolving cracks , 2004 .

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

[20]  T. Belytschko,et al.  A new implementation of the element free Galerkin method , 1994 .

[21]  Guirong Liu,et al.  A point interpolation method for two-dimensional solids , 2001 .

[22]  Carsten Franke,et al.  Solving partial differential equations by collocation using radial basis functions , 1998, Appl. Math. Comput..

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

[24]  Manuel Doblaré,et al.  On solving large strain hyperelastic problems with the natural element method , 2005 .

[25]  R. Plunkett,et al.  Formulas for Stress and Strain , 1965 .

[26]  Satya N. Atluri,et al.  A modified collocation method and a penalty formulation for enforcing the essential boundary conditions in the element free Galerkin method , 1998 .

[27]  Kai Hormann,et al.  Barycentric coordinates for arbitrary polygons in the plane , 2005 .

[28]  Hehua Zhu,et al.  A GENERALIZED AND EFFICIENT METHOD FOR FINITE COVER GENERATION IN THE NUMERICAL MANIFOLD METHOD , 2013 .

[29]  Xuhai Tang,et al.  A novel twice-interpolation finite element method for solid mechanics problems , 2010 .

[30]  Hung Nguyen-Xuan,et al.  A node-based smoothed finite element method (NS-FEM) for upper bound solution to visco-elastoplastic analyses of solids using triangular and tetrahedral meshes , 2010 .

[31]  Wing Kam Liu,et al.  Reproducing kernel particle methods , 1995 .

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

[33]  Charles E. Augarde,et al.  A new partition of unity finite element free from the linear dependence problem and possessing the delta property , 2010 .

[34]  Guirong Liu,et al.  A LOCAL RADIAL POINT INTERPOLATION METHOD (LRPIM) FOR FREE VIBRATION ANALYSES OF 2-D SOLIDS , 2001 .

[35]  Genki Yagawa,et al.  Linear dependence problems of partition of unity-based generalized FEMs , 2006 .

[36]  K. Bathe,et al.  Effects of element distortions on the performance of isoparametric elements , 1993 .

[37]  M. Floater.  Mean value coordinates , 2003, Computer Aided Geometric Design.

[38]  Guirong Liu,et al.  A meshfree radial point interpolation method (RPIM) for three-dimensional solids , 2005 .

[39]  Xuhai Tang,et al.  A novel virtual node method for polygonal elements , 2009 .

[40]  R. Franke.  Scattered data interpolation: tests of some methods , 1982 .

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

[42]  Hong Zheng,et al.  A three-node triangular element with continuous nodal stress , 2014 .

[43]  Zhu Hehua.  Construction of physical cover approximation in manifold method based on least square interpolation , 2009 .

[44]  Zhu Hehua,et al.  A meshless local natural neighbour interpolation method for stress analysis of solids , 2004 .

[45]  B. Nayroles,et al.  Generalizing the finite element method: Diffuse approximation and diffuse elements , 1992 .

[46]  L. Lucy.  A numerical approach to the testing of the fission hypothesis. , 1977 .

[47]  R. Taylor.  The Finite Element Method, the Basis , 2000 .

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

[49]  H. Nguyen-Xuan,et al.  Assessment of smoothed point interpolation methods for elastic mechanics , 2010 .

[50]  P. A. Sackinger,et al.  A finite element method for free surface flows of incompressible fluids in three dimensions. Part I. Boundary fitted mesh motion , 2000 .

[51]  Robert W. Zimmerman,et al.  An impulse-based energy tracking method for collision resolution , 2014 .

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

[53]  Baili Zhang,et al.  A “FE-meshfree” QUAD4 element based on partition of unity , 2007 .

[54]  J. Hoschek,et al.  Scattered Data Interpolation , 1992 .

[55]  Hung Nguyen-Xuan,et al.  Computation of limit and shakedown loads using a node‐based smoothed finite element method , 2012 .

[56]  Shaofan Li,et al.  Meshfree simulations of plugging failures in high-speed impacts , 2010 .

[57]  Hung Nguyen-Xuan,et al.  An alternative alpha finite element method with discrete shear gap technique for analysis of laminated composite plates , 2011, Appl. Math. Comput..

[58]  Baili Zhang,et al.  ‘FE-Meshfree’ QUAD4 element for free-vibration analysis , 2008 .

[59]  J. Monaghan,et al.  Smoothed particle hydrodynamics: Theory and application to non-spherical stars , 1977 .

[60]  Stéphane Bordas,et al.  Strain smoothing in FEM and XFEM , 2010 .

[61]  L. C. Wrobel,et al.  Properties of Gaussian radial basis functions in the dual reciprocity boundary element method , 1993 .

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