Simulation of fracture processes using global-local approach and virtual elements

The underlying Habilitation aims to contribute to the research on fracture mechanics of solids across the scales. This active research field is driven by the investigation and development of new methods, processes and technologies applicable to engineering problems with complex material behavior of solids at fracture. It includes mathematically precise formulations of theoretical and computational models with emphasis on continuum physics as well as the development of variation methods and efficient numerical implementations tools. In particular, two directions will be considered in this contribution: (i) the construction of advanced multi-scale techniques and (ii) modern element technologies. On the multi-scale techniques, a robust and efficient Global-Local approach for numerically solving fracture-mechanics problems is developed in the first part of this contribution. This method has the potential to tackle practical field problems in which a large-structure might be considered and fracture propagation is a localized phenomenum. In this regard, failure is analyzed on a lower (Local) scale, while dealing with a purely linear problem on an upper (Global) scale. The modeling of crack formation at the Local scale is achieved in a convenient way by continuum phase-field formulations to fracture, which are based on the regularization of sharp crack discontinuities. For this purpose, a predictor-corrector scheme is designed in which the local domains are dynamically updated during the computation. To cope with different element discretizations at the interface between the two nested scales, a non-matching dual mortar method is formulated. Hence, more regularity is achieved on the interface. The development of advanced discretization schemes accounting for meshes with highly irregular shaped elements and arbitrary number of nodes is the main focus in the second part of this work. To this end, a relatively new method - the virtual element method (VEM) - will be presented here that leads to an exceptional efficient and stable formulation for solving a wide range of boundary value problems in science and engineering. The structure of VEM comprises a term in the weak formulation or the potential density functional in which the unknowns, being sought are replaced by their projection onto a polynomial space. This results in a rank-deficient structure, therefore it is necessary to add a stabilization term to the formulation. The performance of the virtual element method is comparable to using finite elements of higher order. It is even more robust than FEM in case of a severe distortion of the element