Locking-free shells with structure-preserving mixed finite elements
Date:
Upcoming: The slides will be provided soon.
Abstract:
Nonlinear shell models couple membrane, bending, and, for Reissner-Mindlin type theories, transverse shear effects. Their discretization is challenging because standard finite element methods typically suffer from membrane and shear locking. At the same time, the bending energy involves curvature quantities that are only well-defined on differentiable surfaces. In this talk, we present a structure-preserving finite element framework for nonlinear shells that combines ideas from mixed finite elements and discrete differential geometry.
For the bending energy, we introduce a generalized Weingarten tensor that yields a weak notion of curvature on discretized surfaces, including affine triangulations. The construction is based on a mixed formulation with the bending moment tensor as an additional unknown, leading to methods in the spirit of the Hellan-Herrmann-Johnson approach. This avoids the need for C1-conforming elements and provides a consistent discretization of shell bending on nonsmooth geometries.
To address membrane locking, we show that the membrane strains should be projected into Regge finite element spaces, whose tangential-tangential continuity makes them the natural discrete setting for strain and metric quantities. For Reissner-Mindlin shells, shear locking is avoided by a hierarchical extension using tangentially continuous Nedelec elements, analogous to the TDNNS method. Several examples implemented in the open-source finite element software NGSolve (www.ngsolve.org) demonstrate the excellent performance of the proposed shell elements.