Conforming polygonal finite elements

In this paper, conforming finite elements on polygon meshes are developed. Polygonal finite elements provide greater flexibility in mesh generation and are better‐suited for applications in solid mechanics which involve a significant change in the topology of the material domain. In this study, recent advances in meshfree approximations, computational geometry, and computer graphics are used to construct different trial and test approximations on polygonal elements. A particular and notable contribution is the use of meshfree (natural‐neighbour, nn) basis functions on a canonical element combined with an affine map to construct conforming approximations on convex polygons. This numerical formulation enables the construction of conforming approximation on n‐gons (n⩾3), and hence extends the potential applications of finite elements to convex polygons of arbitrary order. Numerical experiments on second‐order elliptic boundary‐value problems are presented to demonstrate the accuracy and convergence of the proposed method. Copyright © 2004 John Wiley & Sons, Ltd.

[1]  Alireza Tabarraei,et al.  Adaptive computations on conforming quadtree meshes , 2005 .

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

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

[4]  Miguel Ángel Martínez,et al.  Overview and recent advances in natural neighbour galerkin methods , 2003 .

[5]  John E. Bolander,et al.  Numerical Computation of Discrete Differential Operators on Non-Uniform Grids , 2003 .

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

[7]  Eugenio Oñate,et al.  The meshless finite element method , 2003 .

[8]  Eugenio Oñate,et al.  Polyhedrization of an arbitrary 3D point set , 2003 .

[9]  S. Wandzurat,et al.  Symmetric quadrature rules on a triangle , 2003 .

[10]  Alejandro R. Diaz,et al.  Designing materials with prescribed elastic properties using polygonal cells , 2003 .

[11]  N. Sukumar,et al.  Voronoi cell finite difference method for the diffusion operator on arbitrary unstructured grids , 2003 .

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

[13]  Michael S. Floater,et al.  One-to-one piecewise linear mappings over triangulations , 2003, Math. Comput..

[14]  Michael S. Floater,et al.  Mean value coordinates , 2003, Comput. Aided Geom. Des..

[15]  John F. Peters,et al.  Application of the 2-D constant strain assumption to FEM elements consisting of an arbitrary number of nodes , 2003 .

[16]  Gautam Dasgupta,et al.  Interpolants within Convex Polygons: Wachspress' Shape Functions , 2003 .

[17]  Gautam Dasgupta,et al.  Integration within Polygonal Finite Elements , 2003 .

[18]  Elías Cueto,et al.  Modelling three‐dimensional piece‐wise homogeneous domains using the α‐shape‐based natural element method , 2002 .

[19]  Kokichi Sugihara,et al.  Improving continuity of Voronoi-based interpolation over Delaunay spheres , 2002, Comput. Geom..

[20]  Mark Meyer,et al.  Generalized Barycentric Coordinates on Irregular Polygons , 2002, J. Graphics, GPU, & Game Tools.

[21]  Wing Kam Liu,et al.  Meshfree and particle methods and their applications , 2002 .

[22]  B. Moran,et al.  Natural neighbour Galerkin methods , 2001 .

[23]  B.,et al.  Natural Neighbor Galerkin Methods , 2001 .

[24]  G. Dasgupta,et al.  Shape functions for concave quadrilaterals , 2001 .

[25]  P. M. Gullett,et al.  On a finite element method with variable element topology , 2000 .

[26]  Wolfgang Dahmen,et al.  On Wachspress quadrilateral patches , 2000, Comput. Aided Geom. Des..

[27]  Clark R. Dohrmann,et al.  A method for connecting dissimilar finite element meshes in two dimensions , 2000 .

[28]  Kokichi Sugihara,et al.  Two Generalizations of an Interpolant Based on Voronoi Diagrams , 1999, Int. J. Shape Model..

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

[30]  V. D. Ivanov,et al.  The non-Sibsonian interpolation : A new method of interpolation of the values of a function on an arbitrary set of points , 1997 .

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

[32]  Joe D. Warren,et al.  Barycentric coordinates for convex polytopes , 1996, Adv. Comput. Math..

[33]  Jean Braun,et al.  A numerical method for solving partial differential equations on highly irregular evolving grids , 1995, Nature.

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

[35]  Somnath Ghosh,et al.  Elastic-plastic analysis of arbitrary heterogeneous materials with the Voronoi Cell finite element method , 1995 .

[36]  J. Gout.  Rational Wachspress-type Finite Elements on Regular Hexagons , 1985 .

[37]  Charles A. Hall,et al.  The maximum principle for bilinear elements , 1984 .

[38]  Norman H. Christ,et al.  Weights of links and plaquettes in a random lattice , 1982 .

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

[40]  R. Sibson.  A vector identity for the Dirichlet tessellation , 1980, Mathematical Proceedings of the Cambridge Philosophical Society.

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

[42]  Bruce M. Irons,et al.  EXPERIENCE WITH THE PATCH TEST FOR CONVERGENCE OF FINITE ELEMENTS , 1972 .

[43]  A. G. Greenhill,et al.  Handbook of Mathematical Functions with Formulas, Graphs, , 1971 .

[44]  N. S. Barnett,et al.  Private communication , 1969 .

[45]  David M. Miller,et al.  Handbook of Mathematical Functions With Formulas, Graphs and Mathematical Tables (National Bureau of Standards Applied Mathematics Series No. 55) , 1965 .

[46]  H. Coxeter,et al.  Introduction to Geometry , 1964, The Mathematical Gazette.