The TDNNS and Hellan–Herrmann–Johnson method for nonlinear shells
Date:
The slides can be found here.
Abstract:
Recently the Hellan-Herrmann-Johnson (HHJ) method has been extended from linear Kirchhoff–Love plates to nonlinear Koiter shells [M. Neunteufel and J. Schöberl, The Hellan–Herrmann–Johnson method for nonlinear shells, Computers & Structures, 225, pp. 106109, 2019]. Due to the usage of the bending moment tensor as additional field the fourth order problem is rewritten as a second order mixed saddle-point problem avoiding the necessity of \(C^1\)-conforming finite elements. By hybridization a minimization problem is retained being equivalent to the famous Morley element in the lowest order case. The problem of membrane locking for shells with small thicknesses can be mitigated by the usage of an interpolation operator into the matrix valued tangential-tangential continuous Regge finite elements [M. Neunteufel and J. Schöberl, Avoiding membrane locking with Regge interpolation, Computer Methods in Applied Mechanics and Engineering, 373, pp. 113524, 2021].
It is known that for Reissner–Mindlin plates so-called shear locking occurs. By the usage of TDNNS elements, where the rotations are discretized by H(curl)-conforming Nédélec elements and the bending moment tensor is introduced as additional unknown, this behavior is cured [A. Pechstein and J. Schöberl, The TDNNS method for Reissner–Mindlin plates, Numerische Mathematik, 137(3), pp. 713–740, 2017].
In this talk we extend the TDNNS method to nonlinear Naghdi shells by an hierarchical approach on top of the HHJ Koiter shell model, such that membrane and shear locking is circumvented. The relationship between the models and their linearizations are discussed. We present several numerical examples implemented in the finite element software NGSolve (www.ngsolve.org) confirming the excellent performance of the introduced methods.