Adaptive finite element analysis of geometrically non‐linear plates and shells, especially buckling

Based on both moderate and finite rotation bending theories of thin elastic shells including shear deformation, adaptive non-linear static finite element analysis is treated within a displacement approach and h-adaptivity. The a posteriori error indicator given by Rheinboldt, gained by linearization, is investigated in order to decide whether the deformations influence the indicator explicitely and how parameter dependent problems (like the Reissner–Mindlin model) behave in the process of adaptation. In order to achieve overall consistency, dimensional adaptivity (to 3-D elasticity) is implemented within disturbed subdomains, especially at supports. Results are that Rheinboldt's error indicator is valid under certain restrictions but not directly at bifurcation points and that robustness is not improved by adaptation. Nested quadrilateral finite elements are used for studying pre- and post-buckling states of plates and shells.