High order finite elements for three-dimensional, thin walled nonlinear continua

This thesis presents a strictly three-dimensional high order solid finite element formulation far curved thin and thick-walled, physically nonlinear structures. The implementation is based on a hexahedral element, allowing for an anisotropic Ansatz of the displacement field. The polynomial degree of each separate component of the displacement field can be chosen individually and may also be varied in the three local directions of the element. This anisotropie Ansatz allows the efficient computation of threedimensional plate and shell-like structures. Two different nonlinear material models, the deformation theory and the flow theory of plasticity, are considered and several numerical examples are presented comparing the results of the p-version approach to state-of-the-art h-version approximations. In addition, an alternative to the strictly three-dimensional discretization of the whole computational domain is presented. The hp-d method is applied to locally enhance a two-dimensional finite element computation with a three-dimensional Ansatz in those areas where the dimensionally reduced approximation differs significantly from the exact three-dimensional solution. In terms of computation, this approach is therefore a less demanding technique which fills the gap between two and three-dimensional finite element approximations. In order to quantify the discretization error of p-version approximations an explicit error estimator is applied. This error estimator is investigated for two different model problems and is shown to be reliable if appropriate meshes accounting for singularities in the exact solution are constructed.