ABSTRACT
Shells/plates are widely considered in engineering applications. The corresponding discretization procedures are not yet sufficiently reliable, in particular as regards to shell structures. A major cause of these difficulties lies in the numerical locking phenomena that arise in such formulations, see Hughes & Hinton (1986), Kardestuncer & Norrie (1987), Taylor & Zienkiewicz (2000). It is extremely difficult to obtain a shell finite element that is optimal. In the formulation we should aim to satisfy (Bathe et al. 2000, Gilewski 2005): Ellipticity. This condition ensures that the finite element model is solvable and physically means that there are no spurious zero-energy modes.This condition can easily be verified by studying the zero eigenvalues and corresponding eigenvectors of the stiffness matrix of a single unsupported finite element (see for example Gilewski 2005). Consistency. The finite element solution must converge to the solution of the mathematical problem with the element size h close to zero. The bilinear forms used in the finite element discretization must approach the exact bilinear forms of the mathematical model as h approaches zero, Gilewski (1993). Inf-sup condition. Satisfying this condition implies uniform and optimal convergence in bendingdominated shell problems.The shell element is released from shear and membrane locking with solution accuracy independent of the shell thickness parameter. In general, it is very difficult to prove analytically whether a shell or plate finite element satisfies this condition and numerical tests are to be employed.
