The Hellan–Herrmann–Johnson Method for Nonlinear Shells
Date:
The slides can be found here.
Abstract:
Finding appropriate discretizations for nonlinear shells is still a challenging problem. For Kirchhoff plates the Hellan-Herrmann-Johnson method introduces a moment tensor for computing the fourth order equation as a mixed method. The TDNNS method can be seen as an extension to Reissner-Mindlin plates.
In this talk we present a generalization of these methods to nonlinear shells, where we allow large strains and rotations. We may assume the Kirchhoff-Love hypothesis to neglect shearing terms and focus on the bending energy, which is defined as the difference between the curvatures of the deformed and undeformed configuration of the shell. Therefore, we first show how the Weingarten tensor can be computed in a variational sense using H(divdiv)-conforming finite elements. Then we introduce the moment tensor as the Lagrange parameter of the curvatures.
In the shearing case, we use H(curl)-conforming Nédélec elements for additional rotation degrees of freedom. With these elements, also non-smooth surfaces with kinks can be handled directly without rewriting terms.
The method is implemented in NGS-Py, which is based on the finite element library Netgen/NGSolve (www.ngsolve.org). Finally, we present numerical results.