Stability of Some Finite Element Methods for Finite Elasticity Problems

We consider the finite elasticity problem for incompressible materials, proposing a simple bidimensional problem for which we provide indications on the solution stability. Furthermore, we study the stability of the discrete solution, obtained by means of some well-known finite elements, and we present several numerical experiments in order to evaluate and compare the performance of the different discrete schemes under investigation on the considered model problem.

[1]  J. C. Simo,et al.  A CLASS OF MIXED ASSUMED STRAIN METHODS AND THE METHOD OF INCOMPATIBLE MODES , 1990 .

[2]  Alessandro Reali,et al.  An analysis of some mixed-enhanced finite element for plane linear elasticity , 2005 .

[3]  Peter Wriggers,et al.  A note on enhanced strain methods for large deformations , 1996 .

[4]  F. Auricchio,et al.  On the enhanced strain technique for elasticity problems , 2003 .

[5]  Carlo Lovadina,et al.  Analysis of strain‐pressure finite element methods for the Stokes problem , 1997 .

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

[7]  Robert L. Taylor,et al.  FEAP - - A Finite Element Analysis Program , 2011 .

[8]  Michel Fortin,et al.  Mixed and Hybrid Finite Element Methods , 2011, Springer Series in Computational Mathematics.

[9]  Stefanie Reese,et al.  A new stabilization technique for finite elements in non-linear elasticity , 1999 .

[10]  P. Tallec.  Numerical methods for nonlinear three-dimensional elasticity , 1994 .

[11]  M. Chipot.  Finite Element Methods for Elliptic Problems , 2000 .

[12]  Klaus-Jürgen Bathe,et al.  A FOUR-NODE QUADRILATERAL MIXED-INTERPOLATED ELEMENT FOR SOLIDS AND FLUIDS , 1995 .

[13]  Peter Wriggers,et al.  A stabilization technique to avoid hourglassing in finite elasticity , 2000 .

[14]  J. C. Nagtegaal,et al.  Using assumed enhanced strain elements for large compressive deformation , 1996 .

[15]  Yong Liu,et al.  Higher order stabilized finite element method for hyperelastic finite deformation , 2002 .

[16]  P. Tallec.  Existence and approximation results for nonlinear mixed problems: Application to incompressible finite elasticity , 1982 .

[17]  D. Braess.  Enhanced assumed strain elements and locking in membrane problems , 1998 .

[18]  M. Shephard,et al.  A stabilized mixed finite element method for finite elasticity.: Formulation for linear displacement and pressure interpolation , 1999 .

[19]  Klaus-Jürgen Bathe,et al.  On the stability of mixed finite elements in large strain analysis of incompressible solids , 1997 .

[20]  Alessandro Reali,et al.  A stability study of some mixed finite elements for large deformation elasticity problems , 2005 .

[21]  Francisco Armero,et al.  On the locking and stability of finite elements in finite deformation plane strain problems , 2000 .

[22]  J. C. Simo,et al.  Geometrically non‐linear enhanced strain mixed methods and the method of incompatible modes , 1992 .

[23]  J. C. Simo,et al.  Stability and convergence of a class of enhanced strain methods , 1995 .

[24]  M. Fortin,et al.  A stable finite element for the stokes equations , 1984 .

[25]  Francisco Armero,et al.  On the formulation of enhanced strain finite elements in finite deformations , 1997 .

[26]  R. D. Wood,et al.  Nonlinear Continuum Mechanics for Finite Element Analysis , 1997 .