[For the entire collection see Zbl 0604.00019.]
We show how to approximate stresses and free vibration modes in a general thin shell by making use of the following three concepts:
(i) The linear thin shell theory of Koiter which allows us to state the problem in a plane domain \(\Omega \subset {\mathbb{R}}^ 2\). Components of displacement field of middle surface of the shell are the unknowns. Corresponding system of partial differential equations is of order 2 with respect to local tangential components, and of order 4 with respect to local normal components of displacement.
(ii) Some conforming finite element methods. The normal component or the displacement is approximated by using a discrete space constructed from triangles of Argyris or Hsieh-Clough-Tocher which are either complete or reduced. The approximation of the tangential components uses the same spaces, or, other spaces constructed from Lagrange triangles of type 1 or 2. Of course, we employ numerical integration techniques.
(iii) The numerical analysis of thin shell problems of the first author and P. G. Ciarlet [Comput. Meth. Appl. Sci. Eng., 2nd int. Symp., Versailles 1975, Lect. Notes Econ. Math. Syst. 134, 89-136 (1976; Zbl 0356.73066)] and the first author [e.g. Int. J. Eng. Sci. 18, 249-276 (1980; Zbl 0429.73084)]. These results prove the convergence and give asymptotic error estimates. In particular, for the approximation of the displacements, sufficient conditions define the degree of accuracy of the numerical quadrature schemes.