Polygonal Reissner-Mindlin shell element formulation

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The development and analysis of numerical methods for the approximation of the solution to partial differential equations on polygonal meshes have undergone an explosive interest in recent years among the scientific community. In the research project, we want to exploit the inherent advantages of polygonal meshes for shell analysis. Polygonal meshes offer a very flexible framework to handle hanging nodes and different element shapes within the same mesh. Our main objective is to develop a computational framework that makes direct use of the polygonal mesh surface modeling technique. In the case of thin structures, we want to directly employ this mesh as a starting point for the computational analysis. Our primary goal is to develop a polygonal shell element formulation which is accurate, efficient, and allowing for large load step size. To achieve this, three objectives are defined. The first objective is the development of a polygonal shell finite element formulation based on the Reissner-Mindlin theory for the analysis of a wide class of non-linear structural mechanics problems. Methods will be developed to alleviate locking effects for low order polygonal elements. Our goal is a stable element without zero energy modes allowing for large load steps in the non-linear analysis. The quality of the polygonal element employing Voronoi and quadtree meshes will be investigated. The second objective is to exploit the various possibilities in mesh generation and refinement, which are provided by the polygonal element formulation. These are, highly localized mesh refinement, handling of hanging nodes, aligning element boundaries to the domain of interest, and simple remeshing of evolving domains. These features will be exploited to simulate brittle fracture in Reissner-Mindlin shells by employing a phase-field model. The third objective is a NURBS based polygonal shell formulation to exploit higher order continuity. The usage of B-spline polygonal patches offers the possibility to apply k-refinement and local refinement with hierarchical B-splines. The different meshing methods will be analyzed in terms of accuracy and computational cost. The flexibility of the element formulation opens up new possibilities to locally adjust and refine the mesh structure allowing for a tightly interleaved operation of mesh generation and simulation. We aim at methods applicable to a broad class of problems in solid mechanics, including geometric and physically nonlinear problems.