Project Details
Projekt
Foundation and Application of Generalized Mixed FEM Towards Nonlinear Problems in Solid Mechanics
Applicant
Professor Dr. Carsten Carstensen
Subject Area
Mechanics
Term
from 2014 to 2024
Project identifier
Deutsche Forschungsgemeinschaft (DFG) - Project number 255510958
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. The practical applications in computational engineering will be the focus of the Workgroup LUH at the Leibniz University Hannover in cooperation with the Workgroup HU at the Humboldt Universität 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.The focus of the second funding period will be a further close collaboration of both workgroups regarding the extension of the dPG FEM to nonlinear-elastic material behaviour. Therefore, various dPG formulations will be investigated and exercised on relevant mechanical problems by the Workgroup LUH. The implementation in AceGen facilitates expeditious and efficient comparison among different discretizations with respect to convergence behaviour of this novel finite element formulations. The workgroup HU will continue their analysis on nonlinear problems in raising difficulty from Hencky material to polyconvex material and geometric nonlinear configurations. Recent breakthroughs in the dPG methodology for nonlinear problems motivate the application of adaptive dPG schemes with built-in error control to further problems such as hyperelasticity, the obstacle problem and time-evolving elastoplasticity. Optimal convergence rates of adaptive nonlinear LS and dPG methods and Arnold-Winther FEM and guaranteed error estimation for dPG methods involving explicit constants and correct scaling will be investigated.Outreach activities such as coorganization of minisymposia or Oberwolfach workshops (e.g. »Computational Engineering« in 2015, 2018) have fostered the exchange of ideas and fruitful collaborations within the SPP and beyond.
DFG Programme
Priority Programmes
