Distributional curvature approximations with applications to shells
Date:
The slides can be found here.
Abstract:
To compute the Gaussian or mean curvature of a surface embedded in \(\mathbb{R}^3\) in weak sense, a \(C^1\)-surface is required. For a piece-wise affine triangulation this assumption is no longer fulfilled. The approximation of curvature quantities on discrete surfaces is still a field of intensive research in discrete differential geometry (DDG).
The Hellan–Herrmann–Johnson (HHJ) method avoids \(C^1\)-conforming finite elements for the biharmonic plate equation, \(\Delta^2 w=f\), by means of a mixed method discretizing the bending moments, \(\sigma=\nabla^2 w\), by tensor valued elements, where only the normal-normal component is globally continuous [M. Comodi, The Hellan-Herrmann-Johnson method: some new error estimates and postprocessing, Mathematics of Computation, 52 (1989), 17–29]. Regge finite elements, which are matrix-valued and solely tangential-tangential continuous, turned out to be the appropriate space for discretizing strain and metric fields [L. Li, Regge Finite Elements with Applications in Solid Mechanics and Relativity, PhD thesis, University of Minnesota (2018)].
In this talk we combine DDG with HHJ and Regge finite elements to introduce well-defined (high-order) distributional curvature quantities and discuss their convergence [J. Gopalakrishnan, M. Neunteufel, J. Schöberl, M. Wardetzky, Analysis of curvature approximations via covariant curl and incompatibility for Regge metrics, arXiv:2206.09343]. We apply this framework to nonlinear Koiter and Naghdi shells for the bending energy [M. Neunteufel, J. Schöberl, The Hellan–Herrmann–Johnson method for nonlinear shells, Computers & Structures, 225 (2019), 106109] and show how the problem of membrane locking can be mitigated [M. Neunteufel, J. Schöberl, Avoiding membrane locking with Regge interpolation, Comput. Methods Appl. Mech. Eng., 373 (2021), 113524]. We demonstrate the performance of the method by means of benchmark examples implemented in the finite element software NGSolve (www.ngsolve.org).