A discrete geometric view on shear-deformable shell models

Abstract

This thesis presents the construction of a geometrically nonlinear shear-deformable (Cosserat type) shell model by methods from discrete differential geometry (DDG). The model aims at applications in real-time simulations of large deformations of plates and shells. While the more classical finite element approach has shown to yield quantitatively accurate models for a high number of degrees of freedom, the framework of DDG, originally used in computer graphic applications to construct simple yet physically plausible simulations, allows to obtain a concise geometric model that behaves qualitatively correct already on very coarse meshes. The described Discrete Cosserat Shell model is built in several steps. The smooth deformation energy of a shearable shell, usually expressed in an index-prone coordinate notation adapted to a finite element discretization, is reformulated in terms of few, frame-invariant, differential geometric entities. To obtain a discrete deformation energy, we then use insights from the solution theory for the linear Reissner-Mindlin plate equations to construct discrete pendants of these entities which preserve the core properties of the smooth ones. The resulting deformation energy is proven to be consistent with the smooth energy. Numerical validations on linear and nonlinear benchmarks further illustrate the good qualitative behavior of the model as well as its satisfying convergence behavior.