A finite element model for the analysis of the nonlinear mechanical behavior of hybrid composite materials

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The research project deals with a non-standard discretization method. It is based on the fundamental paradigm of scaling the boundary surfaces with respect to a center. It leads to a finite element mesh with star-convex elements that allow hanging nodes, reentrant corners and curved edges. For the resulting polygon or polyhedron-like structures, the corresponding element formulations are developed. In order to enable a general application, these are designed for geometrical and physically non-linear 2D and 3D problems. We are aiming for an element formulation, which combines the advantages of the finite element method and the geometrically exact description of the boundary geometry. The element allows for an arbitrary number of sides. Straight edges or plane surfaces are possible as well as curved edges or surfaces, which are described by e.g. non-uniform rational B splines. In combination with the fast Quadtree or Octree algorithm and a local refinement strategy, it leads to a recursive discretization method. Initially, it starts with the boundary representation of a solid and then automatically continues with a block partitioning of the structure. In this manner, internal interfaces originating from different materials or voids are resolved, and the curved boundary is embedded in the finite element mesh.At first, a Quadtree or Octree decomposition of the domain into star-convex subdomains or elements provides a rough initial discretization. Further mesh refinements are needed to reduce the approximation error of the stresses and displacements. There are two refinements options. One recursively applies Quadtree or Octree decomposition to create the next finer level of discretization. The other considers the parameterization on the element, where the refinement applies either in circumferential direction along the boundary or in radial direction. Note that this step may be completely restricted to a subdomain or the element under consideration. It is not necessary to refine neighboring elements due to so-called hanging nodes. Therefore, it is a local refinement, which can be applied in different ways. It is possible to introduce additional nodes at the boundary edges and/or in the interior of an element, or it is possible to increase the polynomial degree at element level. The different refinement strategies need to be evaluated based on meaningful criteria.The discretization concept with the embedded element formulation results in a general numerical finite element method, which is suitable for the analysis of heterogeneous materials with inclusions and voids. The aim of the project is to contribute to reliable and robust numerical analysis to meet the requirements for the design of modern hybrid composites.