AN INTERPOLATION THEORY APPROACH TO SHELL EIGENVALUE PROBLEMS

The asymptotic behavior of the smallest eigenvalue in linear shell problems is studied, as the thickness parameter tends to zero. In order to cover the widest range of mid-surface geometry and boundary conditions, an abstract approach has been followed, and the Real Interpolation Theory has been used as main tool. A result concerning the ratio between the bending and the total elastic energy is proved. Furthermore, an example of application to cylindrical shells is detailed.

[1]  J. Lions,et al.  Sur Une Classe D’Espaces D’Interpolation , 1964 .

[2]  D. Griffin,et al.  Finite-Element Analysis , 1975 .

[3]  E. Sanchez-Palencia,et al.  Statique et dynamique des coques minces. II: Cas de flexion pure inhibée. Approximation membranaire , 1989 .

[4]  Philippe G. Ciarlet,et al.  Introduction to Linear Shell Theory , 1989 .

[5]  E. Sanchez-Palencia,et al.  Asymptotic and spectral properties of a class of singular-stiff problems , 1992 .

[6]  J. Pitkäranta.  The problem of membrane locking in finite element analysis of cylindrical shells , 1992 .

[7]  D. Chenais,et al.  On the locking phenomenon for a class of elliptic problems , 1994 .

[8]  Juhani Pitkäranta,et al.  On the membrane locking of h-p finite elements in a cylindrical shell problem , 1994 .

[9]  Otso Ovaskainen,et al.  Shell deformation states and the finite element method : a benchmark study of cylindrical shells , 1995 .

[10]  Douglas N. Arnold,et al.  Locking-free finite element methods for shells , 1997, Math. Comput..

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

[12]  D. Chapelle,et al.  Stabilized Finite Element Formulations for Shells in a Bending Dominated State , 1998 .

[13]  Adel Blouza,et al.  Existence and uniqueness for the linear Koiter model for shells with little regularity , 1999 .

[14]  J. Pitkäranta,et al.  A benchmark study of reduced-strain shell finite elements: quadratic schemes , 2000 .

[15]  Alexander G Iosilevich,et al.  An inf-sup test for shell finite elements , 2000 .

[16]  Alexander G Iosilevich,et al.  An evaluation of the MITC shell elements , 2000 .

[17]  K. Bathe,et al.  Optimal consistency errors for general shell elements , 2001 .

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

[19]  Ville Havu,et al.  Analysis of a bilinear finite element for shallow shells I: Approximation of inextensional deformations , 2001, Math. Comput..

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

[21]  Ferdinando Auricchio,et al.  Remarks on the asymptotic behaviour of Koiter shells , 2002 .

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

[23]  Dominique Chapelle,et al.  A shell problem ‘highly sensitive’ to thickness changes , 2003 .

[24]  Ville Havu,et al.  Analysis of a bilinear finite element for shallow shells. II: Consistency error , 2003, Math. Comput..

[25]  D. Chapelle,et al.  MITC elements for a classical shell model , 2003 .

[26]  Lourenço Beirão da Veiga,et al.  Uniform error estimates for a class of intermediate cylindrical shell problems , 2004, Numerische Mathematik.

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

[28]  A NEW APPROACH TO LINEAR SHELL THEORY , 2005 .

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

[30]  L. B. D. Veiga,et al.  ASYMPTOTIC AND NUMERICAL ANALYSIS OF THE EIGENVALUE PROBLEM FOR A CLAMPED CYLINDRICAL SHELL , 2008 .

[31]  H. Triebel.  Theory of Function Spaces III , 2008 .