Differential geometry and the geometrically non-linear Reissner-Mindlin shell model

Abstract

Dedicated to simulating thin-walled structures using the finite element method, this thesis focuses on a consistent Reissner-Mindlin shell formulation through theoretical and numerical investigations. Emphasizing a robust mathematical foundation, particularly in differential geometry, the work explores aspects such as the derivation of stress resultants, consistent linearization, and properties of director interpolation. A pivotal outcome is a finite element formulation that outperforms existing ones, exhibiting key features like objectivity, adherence to unit length constraints, avoidance of path dependence, singularity prevention, and optimal convergence orders. Notably, the study of the consistent linearization process yields the correct tangent operator, identified as the symmetric Riemannian Hessian, serving as the stiffness matrix. This, combined with the study of the correct update of the nodal directors, contributes to the superior convergence behavior of a Newton-Raphson scheme compared to existing formulations. Addressing the assumption of zero transverse normal stress, the thesis proposes a novel numerical treatment, using optimization on manifolds, applicable to arbitrary material models. This method shows potential applicability to other models with stress constraints. The claim of a physically and algorithmically sound Reissner-Mindlin shell formulation is supported by results from numerical investigations. Beyond contributing to the algorithmic treatment of the Reissner-Mindlin shell model, the proposed procedures may have implications for improving the accuracy, efficiency, and reliability of numerical treatments of other structural models.