PP 1748/1: Foundation and Application of Generalized Mixed FEM Towards Nonlinear Problems in Solid Mechanics

The research of this project aims at the mathematical foundation and the engineering application of generalised mixed FEM as well as the development and the analysis of new non-standard methods that yield guaranteed results for nonlinear problems in solid mechanics. The practical applications in computational engineering is the focus of the Workgroup LUH at the Leibniz University Hannover in cooperation with the Workgroup HU at the Humboldt Universit?t zu Berlin with focus on mathematical foundation of the novel discretization schemes. The joint target is the effective and reliable simulation in nonlinear continuum mechanics with development of adaptive numerical discretizations based on ultraweak formulations between nonconforming, mixed and discontinuous Galerkin or Petrov-Galerkin Finite Element Methods. In the first funding period, the workgroup LUH developed different discontinuous discretization methods. An efficient extension/enhancement of the original discontinuous Galerkin Finite Element Method (dG FEM) avoids shear-locking effects and volumetric-locking for (nearly) incompressible and elasto-plastic material behaviour. The workgroup HU developed and analysed a discontinuous Petrov-Galerkin (dPG) FEM for a nonlinear model problem in collaboration with the workgroup LUH and proved optimal convergence rates of adaptive dPG and least-squares methods for linear elastic problems. Further topics of research were guaranteed error bounds for pointwise symmetric discretizations in linear elasticity and the analysis of nonconforming FEM for polyconvex materials.