DE | EN
Home
About Us
Overview
Facts and Figures
Organization
Scientists
Contact
Approach
Situations offered
Research
Overview
Application Fields
Projects
Publications
Scientists
Preprints
Institutional Cooperation
Archiv 02-14
Transfer
Overview
Industry
References
MODAL-AG
Spin Offs
Software
Patents
Schools
Overview
MathInside
MATHEATHLON
Matheon-Kalender
What'sMath
Training for Teachers
Summer Schools
Events
Press
Overview
Releases
News
Overview
Matheon Head
Number of the week
News 2002 - 2014
Activities
Overview
Workshops
15 Years Matheon
Media
Overview
Photos
Videos
Audios
Booklets
Books
News from around the world

Since 2019, Matheon's application-oriented mathematical research activities are being continued in the framework of the Cluster of Excellence MATH+
www.mathplus.de
The Matheon websites will not be updated anymore.

Friederike Hellwig

hellwigf@math.hu-berlin.de


Projects as a member

  • SE-AP14

    Foundation and application of generalized mixed FEM towards nonlinear problems in solid mechanics

    Prof. Dr. Carsten Carstensen

    Project heads: Prof. Dr. Carsten Carstensen
    Project members: Philipp Bringmann / Friederike Hellwig
    Duration: 01.09.2014 - 30.11.2019
    Status: running
    Located at: Humboldt Universität Berlin

    Description

    Despite the practical success in computational engineering and a few partial mathematical convergence proofs, many fundamental questions on the reliable and effective computer simulation in nonlinear mechanics are still open. The success of mixed FEMs in the linear elasticity with focus on the accuracy of the stress variable motivated the research of novel discretization schemes in the SPP1748. This and recent surprising advantages of related nonconforming finite element methods in nonlinear partial differential equations with guaranteed lower eigenvalue bounds or lower energy bounds in convex minimization problems suggests the investigation of mixed and simpler generalized mixed finite element methods such as discontinuous Petrov-Galerkin schemes for linear or linearized elasticity and nonlinear elasticity with polyconvex energy densities in this project. The practical applications in computational engineering will be the focus of the Workgroup LUH with all 3D simulations to provide numerical insight in the feasibility and robustness of the novel simulation tools, while the Workgroup HU will provide mathematical foundation of the novel schemes with rigorous a priori and a posteriori error estimates. The synergy effects of the two workgroups will be visible in that problems with a known rigorous mathematical analysis or the Lavrentiev gap phenomenon or cavitation will be investigated by engineers for the first time and, vice versa, more practical relevant models in nonlinear mechanics will be looked at from a mathematical viewpoint with arguments from the calculus of variations and the implicit function theorem combined with recent arguments for a posteriori error analysis and adaptive mesh-refining. A combination of ideas in least-squares finite element methods with those of hybridized methods recently led to discontinuous Petrov Galerkin (dPG) FEMs. They minimize a residual inherited from a piecewise ultra weak formulation in a nonstandard localized dual norm. This innovative ansatz will be generalized from Hilbert to Banach spaces to allow the numerical approximation of linearized problems in nonlinear mechanics which leads to some global inf-sup condition on the continuous and on the discrete level for stability of the novel ultra weak formulations. The joint interest is the design of adaptive algorithms for effective mesh-design and the understanding of the weak or penalized coupling of the nonlinear stress-strain relations. A key difficulty arises from the global or localized and then numerical inversion of the nonlinear stress-strain relation in some overall Hu-Washizu-type mixed formulation. While convex energy densities allow a formal inversion of the stress-strain relation via a duality in convex analysis, it contradicts the frame indifference in continuum mechanics. The extension for polyconvex energy densities is only possible for special cases in closed form but has, in general, to be localized and approximated.

    https://www.uni-due.de/spp1748/generalized_mixed_nonlinear_fem.php

Projects as a guest