Virtual and smoothed finite elements: A connection and its application to polygonal/polyhedral finite element methods

Summary We show both theoretically and numerically a connection between the smoothed finite element method (SFEM) and the virtual element method and use this approach to derive stable, cheap and optimally convergent polyhedral FEM. We show that the stiffness matrix computed with one subcell SFEM is identical to the consistency term of the virtual element method, irrespective of the topology of the element, as long as the shape functions vary linearly on the boundary. Using this connection, we propose a new stable approach to strain smoothing for polygonal/polyhedral elements where, instead of using sub-triangulations, we are able to use one single polygonal/polyhedral subcell for each element while maintaining stability. For a similar number of degrees of freedom, the proposed approach is more accurate than the conventional SFEM with triangular subcells. The time to compute the stiffness matrix scales with the O(dofs)1.1 in case of the conventional polygonal FEM, while it scales as O(dofs)0.7 in the proposed approach. The accuracy and the convergence properties of the SFEM are studied with a few benchmark problems in 2D and 3D linear elasticity. Copyright © 2015 John Wiley & Sons, Ltd.

[1]  John A. Evans,et al.  Isogeometric boundary element analysis using unstructured T-splines , 2013 .

[2]  Wolfgang A. Wall,et al.  An accurate, robust, and easy-to-implement method for integration over arbitrary polyhedra: Application to embedded interface methods , 2014, J. Comput. Phys..

[3]  Gianmarco Manzini,et al.  The Mimetic Finite Difference Method for Elliptic Problems , 2014 .

[4]  Glaucio H. Paulino,et al.  Addressing Integration Error for Polygonal Finite Elements Through Polynomial Projections: A Patch Test Connection , 2013, 1307.4423.

[5]  F. Tin-Loi,et al.  Polygon scaled boundary finite elements for crack propagation modelling , 2012 .

[6]  K. Y. Sze,et al.  Four- and eight-node hybrid-Trefftz quadrilateral finite element models for helmholtz problem , 2010 .

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

[8]  Vadim Shapiro,et al.  Adaptively weighted numerical integration over arbitrary domains , 2014, Comput. Math. Appl..

[9]  P. Bar-Yoseph,et al.  Mechanically based models: Adaptive refinement for B‐spline finite element , 2003 .

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

[11]  Hui-Ping Wang,et al.  Some recent improvements in meshfree methods for incompressible finite elasticity boundary value problems with contact , 2000 .

[12]  S. Rajendran,et al.  A technique to develop mesh-distortion immune finite elements , 2010 .

[13]  Guirong Liu,et al.  A novel alpha finite element method (αFEM) for exact solution to mechanics problems using triangular and tetrahedral elements , 2008 .

[14]  Guirong Liu,et al.  A face‐based smoothed finite element method (FS‐FEM) for 3D linear and geometrically non‐linear solid mechanics problems using 4‐node tetrahedral elements , 2009 .

[15]  J. Wolf,et al.  The scaled boundary finite-element method – a primer: derivations , 2000 .

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

[17]  R. L. Harder,et al.  A proposed standard set of problems to test finite element accuracy , 1985 .

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

[19]  Joseph E. Bishop,et al.  A displacement‐based finite element formulation for general polyhedra using harmonic shape functions , 2014 .

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

[21]  T. Hughes,et al.  Isogeometric analysis : CAD, finite elements, NURBS, exact geometry and mesh refinement , 2005 .

[22]  A. Russo,et al.  New perspectives on polygonal and polyhedral finite element methods , 2014 .

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

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

[25]  G. Golub,et al.  Structured inverse eigenvalue problems , 2002, Acta Numerica.

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

[27]  Stéphane Bordas,et al.  On the performance of strain smoothing for quadratic and enriched finite element approximations (XFEM/GFEM/PUFEM) , 2011 .

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

[29]  B. Lévy,et al.  L p Centroidal Voronoi Tessellation and its applications , 2010, SIGGRAPH 2010.

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

[31]  N. Sukumar,et al.  Generalized Gaussian quadrature rules on arbitrary polygons , 2010 .

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

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

[34]  Guirong Liu,et al.  Extended finite element method with edge-based strain smoothing (ESm-XFEM) for linear elastic crack growth , 2012 .

[35]  E. Ooi,et al.  Mesh‐distortion immunity assessment of QUAD8 elements by strong‐form patch tests , 2006 .

[36]  Qing Hua Qin,et al.  Fundamental-solution-based hybrid FEM for plane elasticity with special elements , 2011 .

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

[38]  A. Rajagopal,et al.  Mesh free Galerkin method based on natural neighbors and conformal mapping , 2008 .

[39]  N. Sukumar,et al.  Archives of Computational Methods in Engineering Recent Advances in the Construction of Polygonal Finite Element Interpolants , 2022 .

[40]  Ligang Liu,et al.  Revisiting Optimal Delaunay Triangulation for 3D Graded Mesh Generation , 2014, SIAM J. Sci. Comput..

[41]  Guirong Liu,et al.  A node-based smoothed finite element method (NS-FEM) for upper bound solutions to solid mechanics problems , 2009 .

[42]  Glaucio H. Paulino,et al.  PolyTop: a Matlab implementation of a general topology optimization framework using unstructured polygonal finite element meshes , 2012 .

[43]  Thanh Hai Ong,et al.  Inf-suf stable bES-FEM method for nearly incompressible elasticity , 2013 .

[44]  Glaucio H. Paulino,et al.  On the Virtual Element Method for three-dimensional linear elasticity problems on arbitrary polyhedral meshes , 2014 .

[45]  Dong-Ming Yan,et al.  Efficient computation of clipped Voronoi diagram for mesh generation , 2013, Comput. Aided Des..

[46]  Marc Duflot,et al.  A cell-based smoothed finite element method for three dimensional solid structures , 2012 .

[47]  Lourenço Beirão da Veiga,et al.  Virtual Elements for Linear Elasticity Problems , 2013, SIAM J. Numer. Anal..

[48]  Elena Atroshchenko,et al.  Fundamental solutions and dual boundary element methods for fracture in plane Cosserat elasticity , 2015, Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences.

[49]  R. Quey,et al.  Large-scale 3D random polycrystals for the finite element method: Generation, meshing and remeshing , 2011 .

[50]  F. Brezzi,et al.  Basic principles of Virtual Element Methods , 2013 .

[51]  J. Trevelyan,et al.  An isogeometric boundary element method for elastostatic analysis: 2D implementation aspects , 2013, 1302.5305.

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

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

[54]  Franco Brezzi,et al.  The Hitchhiker's Guide to the Virtual Element Method , 2014 .

[55]  S. Natarajan,et al.  Convergence and accuracy of displacement based finite element formulations over arbitrary polygons: Laplace interpolants, strain smoothing and scaled boundary polygon formulation , 2014 .

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

[57]  Alvise Sommariva,et al.  Gauss-Green cubature and moment computation over arbitrary geometries , 2009, J. Comput. Appl. Math..

[58]  C. Peskin.  The immersed boundary method , 2002, Acta Numerica.

[59]  P. Hansbo.  A new approach to quadrature for finite elements incorporating hourglass control as a special case , 1998 .