A cell-based smoothed finite element method for three dimensional solid structures

This paper extends further the strain smoothing technique in finite elements to 8-noded hexahedral elements (CS-FEM-H8). The idea behind the present method is similar to the cell-based smoothed 4-noded quadrilateral finite elements (CS-FEM-Q4). In CSFEM, the smoothing domains are created based on elements, and each element can be further subdivided into 1 or several smoothing cells. It is observed that: 1) The CS-FEM using a single smoothing cell can produce higher stress accuracy, but insufficient rank and poor displacement accuracy; 2) The CS-FEM using several smoothing cells has proper rank, good displacement accuracy, but lower stress accuracy, especially for nearly incompressible and bending dominant problems. We therefore propose 1) an extension of strain smoothing to 8-noded hexahedral elements and 2) an alternative CS-FEM form, which associates the single smoothing cell issue with multi-smoothing cell one via a stabilization technique. Several numerical examples are provided to show the reliability and accuracy of the present formulation.

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

[2]  Michael A. Puso,et al.  A stabilized nodally integrated tetrahedral , 2006 .

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

[4]  N. S. Ottosen,et al.  Fast and accurate 4‐node quadrilateral , 2004 .

[5]  Trung Nguyen-Thoi,et al.  A stabilized smoothed finite element method for free vibration analysis of Mindlin–Reissner plates , 2009 .

[6]  D. Kelly.  Bounds on discretization error by special reduced integration of the lagrange family of finite elements , 1980 .

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

[8]  Roland Glowinski,et al.  Energy methods in finite element analysis , 1980 .

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

[10]  Clark R. Dohrmann,et al.  A uniform nodal strain tetrahedron with isochoric stabilization , 2009 .

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

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

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

[14]  N. S. Ottosen,et al.  Accurate eight‐node hexahedral element , 2007 .

[15]  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 .

[16]  Guirong Liu,et al.  An edge-based smoothed finite element method for analysis of two-dimensional piezoelectric structures , 2009 .

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

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

[19]  S. Timoshenko,et al.  Theory of elasticity , 1975 .

[20]  Sundararajan Natarajan,et al.  On the approximation in the smoothed finite element method (SFEM) , 2010, ArXiv.

[21]  K. Y. Dai,et al.  Free and forced vibration analysis using the smoothed finite element method (SFEM) , 2007 .

[22]  L. P. Bindeman,et al.  Assumed strain stabilization of the eight node hexahedral element , 1993 .

[23]  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 .

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

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

[26]  L. Richardson.  The Approximate Arithmetical Solution by Finite Differences of Physical Problems Involving Differential Equations, with an Application to the Stresses in a Masonry Dam , 1911 .

[27]  S. Timoshenko,et al.  Theory of Elasticity (3rd ed.) , 1970 .

[28]  Michael A. Puso,et al.  Meshfree and finite element nodal integration methods , 2008 .

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

[30]  Guirong Liu.  A GENERALIZED GRADIENT SMOOTHING TECHNIQUE AND THE SMOOTHED BILINEAR FORM FOR GALERKIN FORMULATION OF A WIDE CLASS OF COMPUTATIONAL METHODS , 2008 .

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

[32]  Stéphane Bordas,et al.  Addressing volumetric locking and instabilities by selective integration in smoothed finite elements , 2009 .

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

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

[35]  J. M. Kennedy,et al.  Hourglass control in linear and nonlinear problems , 1983 .

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

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

[38]  M. Duflot.  A meshless method with enriched weight functions for three‐dimensional crack propagation , 2006 .

[39]  B. D. Veubeke.  Displacement and equilibrium models in the finite element method , 1965 .