Uniform convergence and a posteriori error estimators for the enhanced strain finite element method

Summary.Enhanced strain elements, frequently employed in practice, are known to improve the approximation of standard (non-enhanced) displacement-based elements in finite element computations. The first contribution in this work towards a complete theoretical explanation for this observation is a proof of robust convergence of enhanced element schemes: it is shown that such schemes are locking-free in the incompressible limit, in the sense that the error bound in the a priori estimate is independent of the relevant Lamé constant. The second contribution is a residual-based a posteriori error estimate; the L2 norm of the stress error is estimated by a reliable and efficient estimator that can be computed from the residuals.

[1]  Douglas N. Arnold,et al.  Regular Inversion of the Divergence Operator with Dirichlet Boundary Conditions on a Polygon. , 1987 .

[2]  C. Carstensen,et al.  Constants in Clément-interpolation error and residual based a posteriori estimates in finite element methods , 2000 .

[3]  Carsten Carstensen,et al.  Locking-free adaptive mixed finite element methods in linear elasticity , 2000 .

[4]  L. R. Scott,et al.  The Mathematical Theory of Finite Element Methods , 1994 .

[5]  M. Vogelius.  An analysis of thep-version of the finite element method for nearly incompressible materials , 1983 .

[6]  Ekkehard Ramm,et al.  A class of equivalent enhanced assumed strain and hybrid stress finite elements , 1999 .

[7]  T. R. Hughes,et al.  Mathematical foundations of elasticity , 1982 .

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

[9]  F. Thomasset.  Finite element methods for Navier-Stokes equations , 1980 .

[10]  S. Reese,et al.  A new locking-free brick element technique for large deformation problems in elasticity ☆ , 2000 .

[11]  Rüdiger Verfürth,et al.  A posteriori error estimation and adaptive mesh-refinement techniques , 1994 .

[12]  Byung Chai Lee,et al.  EQUIVALENCE BETWEEN ENHANCED ASSUMED STRAIN METHOD AND ASSUMED STRESS HYBRID METHOD BASED ON THE HELLINGER–REISSNER PRINCIPLE , 1996 .

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

[14]  Robert L. Taylor,et al.  Improved versions of assumed enhanced strain tri-linear elements for 3D finite deformation problems☆ , 1993 .

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

[16]  Juhani Pitkäranta,et al.  An analysis of the discontinuous Galerkin method for a scalar hyperbolic equation , 1986 .

[17]  Michael Vogelius.  An Analysis of the p-Version of the Finite Element Method for Nearly Incompressible Materials Uniformly Valid, Optimal Error Estimates , 1983 .

[18]  P. Clément.  Approximation by finite element functions using local regularization , 1975 .

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

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

[21]  R. Verfürth.  A review of a posteriori error estimation techniques for elasticity problems , 1999 .

[22]  Carsten Carstensen,et al.  A posteriori error estimates for mixed FEM in elasticity , 1998, Numerische Mathematik.

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