On the effects of using curved elements in the approximation of the Reissner--Mindlin plate by the p version of the finite element method

We consider the approximation of the Reissner-Mindlin plate model by the standard Galerkin p version finite element method. Under the assumption of sufficient smoothness on the solution, we illustrate that the method is asymptotically free of locking even when certain curvilinear elements are used. The amount of preasymptotic locking is also identified and is shown to depend on the element mappings. We identify which mappings will result in asymptotically locking free methods and through numerical computations we verify the results for various mappings used in practice.

[1]  I. Babuska,et al.  Locking effects in the finite element approximation of elasticity problems , 1992 .

[2]  I. Babuska,et al.  The problem of plate modeling: theoretical and computational results , 1992 .

[3]  ON THE APPROXIMATION OF THE REISSNER-MINDLIN PLATE BY THE p/hp VERSION OF THE FINITE ELEMENT METHOD , 2002 .

[4]  I. Babuska,et al.  Finite Element Analysis , 2021 .

[5]  Ivo Babuška,et al.  Hierarchic modeling of plates , 1991 .

[6]  D. Arnold,et al.  Asymptotic analysis of the boundary layer for the Reissner-Mindlin plate model , 1996 .

[7]  M. Bernadou.  β1-curved finite elements with numerical integration for thin plate and thin shell problems, Part 2: Approximation of thin plate and thin shell problems , 1993 .

[8]  Mark Ainsworth,et al.  The hp-MITC finite element method for the Reissner-Mindlin plate problem , 2002 .

[9]  I. Babuska,et al.  Locking effects in the finite element approximation of plate models , 1995 .

[10]  Christos Xenophontos.  Finite element computations for the Reissner–Mindlin plate model , 1998 .

[11]  Manil Suri,et al.  Analytical and computational assessment of locking in the hp finite element method , 1996 .

[12]  I. Babuska,et al.  On locking and robustness in the finite element method , 1992 .

[13]  Lawrence K. Chilton,et al.  On the construction of stable curvilinear $p$ version elements for mixed formulations of elasticity and Stokes flow , 2000, Numerische Mathematik.

[14]  Lawrence K. Chilton.  Looking-Free Mixed hp Finite Element Methods for Linear and Geometrically Nonlinear Elasticity , 1997 .

[15]  Manil Suri,et al.  The $p$-version of the finite element method for elliptic equations of order $2l$ , 1990 .

[16]  Christos Xenophontos,et al.  The hp finite element method for problems in mechanics with boundary layers , 1998 .

[17]  D. Arnold,et al.  Edge effects in the Reissner-Mindlin plate theory , 1989 .

[18]  M. Bernadou.  C 1 -curved finite elements with numerical integration for thin plate and thin shell problems, Part 1: construction and interpolation properties of curved C 1 finite elements , 1993 .