Modern finite element methods (FEM) nowadays play an important role in the construction, design and development of new materials, innovative products or production processes. Despite successful research in the past, there are still many open problems, e.g., artificial stiffening effects, numerical instabilities and undesired mesh distortion sensitivity. These deficiencies become especially evident in the presence of non-linearities. Typical examples are geometrical or material non-linearities or do stem from constrained minimization such as incremental plasticity or contact problems.
The discontinuous Galerkin (dG) method and the more recently introduced hybrid versions provide a flexible approach to introduce discontinuities between sub-domains in a variationally consistent way. Being explored mainly from the analytical side, it has not yet found its way into challenging simulations of engineering applications including inelastic and anisotropic material as well complex damage and contact models. To fill this gap, an interdisciplinary group representing three research fields – applied mechanics, numerical analysis and scientific computing – has been formed. In this project a new family of adaptive hybrid discontinuous Galerkin and Petrov-Galerkin schemes will be designed, analyzed and benchmarked. This approach allows us not only to embed well-established highly reliable advanced finite element technology but also to obtain an innovative discretization concept including stress fields as additional primary variables and being robust in the incompressible limit, for finite deformation, and with respect to mesh distortion.