Distributional curvatures on discrete surfaces with application to shells
Date:
The slides can be found here.
Abstract:
To compute the Gaussian or mean curvature of a surface embedded in \(\mathbb{R}^3\) a \(C^1\)-surface is required. For a piece-wise affine triangulation this assumption is no longer fulfilled. The approximation of curvature quantities 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 problem by means of a mixed method discretizing the bending moments 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 strains and metrics [J. Gopalakrishnan, M. Neunteufel, J. Schöberl, M. Wardetzky, Analysis of curvature approximations via covariant curl and incompatibility for Regge metrics, arXiv:2206.09343]. In this talk we combine DDG with HHJ and Regge finite elements to introduce well-defined distributional curvature quantities. We apply this framework to nonlinear 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).