Abstract

The research of this project aims at the mathematical foundation and the engineering application of generalized 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 including finite deformations and nonlinear material. The motivation came from the fact that classical finite elements are not sufficient to overcome locking phenomena. The big difference between linear and nonlinear finite element formulations in the area of incompressibility is that the linear constraint (div u) becomes non-linear (J = detF). In this case many mathematical results cannot be carried over to the finite strain case. This research is based on a strong cooperation between the “Workgroup HU” at the Humboldt-University Berlin and the “Workgroup LUH” at the Leibniz University Hannover. Therefore the practical applications in computational engineering will be the focus of the Workgroup LUH and the Workgroup HU will provide mathematical foundation of the novel discretization schemes. The target is the effective and reliable simulation in nonlinear mechanics with development of adaptive numerical discretizations based on ultraweak formulations between nonconforming, mixed and discontinuous Galerkin or Petrov-Galerkin Finite Element Methods. The different viewpoints shall narrow the gap of what can be theoretically proven in mathematics and what is established numerical experience in mechanics while looking from different angles on the same mathematical and numerical models.