Prof. Dr. Carsten Carstensen

Projects as a project leader

• 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

• SE-AP1

  Numerical algorithms for the simulation of finite plasticity with microstructures

  Prof. Dr. Carsten Carstensen

  Project heads: Prof. Dr. Carsten Carstensen
  Project members: -
  Duration: 01.10.2010 - 30.06.2016
  Status: completed
  Located at: Humboldt Universität Berlin
  

  Description

  The occurrence of microstructures in solid mechanics and, in particular, in finite plasticity can be attributed to a loss of the convexity of the underlying energy potentials. While the material deforms macroscopically, structures in the form of shear bands, cracks or lami- nates arise on microscopic scales. Common to these examples is that their macroscopic simulations have to be based on the quasiconvexification of the energy functional.The projects within the research group either concern the modelling or the simulation. In contrast, the object of this project is the justification of computer simulations with an analysis of discretisation and design of converging adaptive mesh-refining algorithms. The mathematical justification concerns numerical simulations on the microscopic scale (a), on the macroscopic scale (b), for time-evolving microstructure (c).In the first funding period, an efficient algorithm on the numerical relaxation in single- crystal finite plasticity has been established in (a). The degenerate nature of the (quasi-) convexified variational model in (b) required novel stabilisation techniques on adapted finite element grids. The influence of the perturbation of the computed macroscopic energy density W is examined in the combination of (a) and (b). The main result in the analysis of perturbed minimisation problems guarantees convergence of an adaptive mesh-refining algorithm for asymptotically exact computation of energies.The project continues to investigate the convergence of numerical simulations of rate- independent evolution problems for the full time-space discretisation (c). The second funding period shall investigate the improvements of nonstandard finite element methods in (a), (b), and (c).
  
  http://gepris.dfg.de/gepris/projekt/35736987