Adaptive mesh refinements for thin shells whose middle surface is not exactly known

A strategy concerning mesh refinements for thin shells computation is presented. The geometry of the shell is given only by the reduced information consisting in nodes and normals on its middle surface corresponding to a coarse mesh. The new point is that the mesh refinements are defined from several criteria, including the transverse shear forces which do not appear in the mechanical energy of the applied shell formulation. Another important point is to be able to construct the unknown middle surface at each step of the refinement. For this, an interpolation method by edges, coupled with a triangle bisection algorithm, is applied. This strategy is illustrated on various geometries and mechanical problems.

[1]  T. Hughes,et al.  Isogeometric analysis : CAD, finite elements, NURBS, exact geometry and mesh refinement , 2005 .

[2]  Philippe Destuynder,et al.  A mixed finite element for shell model with free edge boundary conditions Part 1. The mixed variational formulation , 1995 .

[3]  J. Z. Zhu,et al.  The superconvergent patch recovery and a posteriori error estimates. Part 1: The recovery technique , 1992 .

[4]  Manfredo P. do Carmo,et al.  Differential geometry of curves and surfaces , 1976 .

[5]  Philippe Destuynder,et al.  A mixed finite element for shell model with free edge boundary conditions Part 2. The numerical scheme , 1995 .

[6]  J. L. Batoz,et al.  Modélisation des Structures par Éléments Finis, par J. L. Batoz et G. Dhatt Volume 1: Solides élastiques; Volume 2: Poutres et plaques; Volume 3: Coques, Editions Hermes, Paris , 1994 .

[7]  J. Tinsley Oden,et al.  Practical methods for a posteriori error estimation in engineering applications , 2003 .

[8]  J. Z. Zhu,et al.  The superconvergent patch recovery and a posteriori error estimates. Part 2: Error estimates and adaptivity , 1992 .

[9]  Roman Lackner,et al.  Mesh generation and mesh refinement procedures for the analysis of concrete shells , 2000 .

[10]  Ekkehard Ramm,et al.  A posteriori error estimation and adaptivity for linear elasticity using the reciprocal theorem , 1998 .

[11]  J. W. Bull,et al.  Adaptive mesh refinement for shells with modified Ahmad elements , 1996 .

[12]  Michel Salaün,et al.  Approximation of shell geometry for non-linear analysis , 1998 .

[13]  R. D. Wood,et al.  Smoothing stress resultants in adaptive finite element shell analysis , 1995 .

[14]  Vivette Girault,et al.  Finite Element Methods for Navier-Stokes Equations - Theory and Algorithms , 1986, Springer Series in Computational Mathematics.

[15]  Philippe G. Ciarlet,et al.  The finite element method for elliptic problems , 2002, Classics in applied mathematics.

[16]  Chung-Souk Han,et al.  An h-adaptive method for elasto-plastic shell problems , 2000 .

[17]  Maenghyo Cho,et al.  r -Adaptive mesh generation for shell finite element analysis , 2004 .

[18]  K. Schweizerhof,et al.  Adaptive mesh generation on arbitrarily curved shell structures , 1997 .

[19]  J. Oden,et al.  A Posteriori Error Estimation in Finite Element Analysis , 2000 .

[20]  W. Flügge.  Stresses in Shells , 1960 .

[21]  M. Rivara.  Algorithms for refining triangular grids suitable for adaptive and multigrid techniques , 1984 .