The MITC9 shell element in plate bending: mathematical analysis of a simplified case

We consider the 9-node shell element referred to as the MITC9 shell element in plate bending solutions and present a simplified mathematical analysis. The element uses bi-quadratic interpolations of the rotations and transverse displacement, and the “rotated Raviart-Thomas” interpolations for the transverse shear stresses. A rigorous mathematical analysis of the element is still lacking, even for the simplified case of plate solutions (that is, flat shells), although the numerical evidence suggests a good and reliable behavior. Here we start such an analysis by considering a very simple particular case; namely, a rectangular plate, clamped all around the boundary, and solved with a uniform decomposition. Moreover, we consider only the so-called limit case, corresponding to the limit equations that are obtained for the thickness t going to zero. While the mathematical analysis of the limit case is simpler, such analysis, in general, gives an excellent indication of whether shear locking is present in the real case t > 0. We detail that the element in the setting considered shows indeed optimal behavior.

[1]  Carlo Lovadina,et al.  A SHELL CLASSIFICATION BY INTERPOLATION , 2002 .

[2]  Phill-Seung Lee,et al.  Insight into finite element shell discretizations by use of the basic shell mathematical model , 2005 .

[3]  Dominique Chapelle,et al.  On the ellipticity condition for model-parameter dependent mixed formulations , 2010 .

[4]  F. Brezzi,et al.  Sur la classification des coques linéairement élastiques , 1999 .

[5]  Lourenço Beirão da Veiga,et al.  Asymptotic energy behavior of two classical intermediate benchmark shell problems , 2002 .

[6]  Klaus-Jürgen Bathe,et al.  On evaluating the inf–sup condition for plate bending elements , 1997 .

[7]  K. Bathe,et al.  Measuring convergence of mixed finite element discretizations: an application to shell structures , 2003 .

[8]  Klaus-Jürgen Bathe,et al.  Displacement and stress convergence of our MITC plate bending elements , 1990 .

[9]  Phill-Seung Lee,et al.  Measuring the convergence behavior of shell analysis schemes , 2011 .

[10]  K. Bathe,et al.  The Mechanics of Solids and Structures - Hierarchical Modeling and the Finite Element Solution , 2011 .

[11]  Jean E. Roberts,et al.  Mixed and hybrid finite element methods , 1987 .

[12]  Phill-Seung Lee,et al.  Towards improving the MITC9 shell element , 2003 .

[13]  L. B. D. Veiga.  Asymptotic study of the solution for pinched cylindrical shells , 2005 .

[14]  K. Bathe,et al.  The MITC7 and MITC9 Plate bending elements , 1989 .

[15]  Phill-Seung Lee,et al.  On the asymptotic behavior of shell structures and the evaluation in finite element solutions , 2002 .

[16]  L. B. D. Veiga,et al.  Numerical evaluation of the asymptotic energy behavior of intermediate shells with application to two classical benchmark tests , 2004 .

[17]  Long Chen.  FINITE ELEMENT METHOD , 2013 .

[18]  Anthony Ralston,et al.  Encyclopedia of computer science and engineering , 1983 .

[19]  M. L. Bucalém,et al.  Higher‐order MITC general shell elements , 1993 .

[20]  Dominique Chapelle,et al.  The Finite Element Analysis of Shells - Fundamentals - Second Edition , 2011 .

[21]  D. Chapelle,et al.  The Finite Element Analysis of Shells - Fundamentals , 2003 .

[22]  Phill-Seung Lee,et al.  The quadratic MITC plate and MITC shell elements in plate bending , 2010, Adv. Eng. Softw..

[23]  H. Kardestuncer,et al.  Finite element handbook , 1987 .

[24]  K. Bathe,et al.  Mixed-interpolated elements for Reissner–Mindlin plates , 1989 .

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

[26]  K. Bathe,et al.  Fundamental considerations for the finite element analysis of shell structures , 1998 .

[27]  M. Fortin,et al.  ERROR ANALYSIS OF MIXED-INTERPOLATED ELEMENTS FOR REISSNER-MINDLIN PLATES , 1991 .

[28]  F. Brezzi,et al.  New Quadrilateral Plate Elements Based on Twist-Kirchhoff Theory , 2010 .