The Hellan-Herrmann-Johnson and TDNNS method for linear and nonlinear shells

TL;DR

The paper develops a locking-free, mixed HHJ/TDNNS framework for nonlinear Koiter and Naghdi shells by introducing a hierarchical shear representation via $H(\mathrm{curl})$-conforming Nédélec elements and a distributional-curvature lifting. Through three-field Hu–Washizu and subsequent reductions, it integrates edge-angle information and hybridization to yield stable, solvable formulations that extend to kinks and branched shells; linearization recovers the classical Kirchhoff–Love and Reissner–Mindlin plate limits with Regge-based locking mitigation. The approach unifies nonlinear shell formulations with robust, high-order discretizations and shows strong numerical performance across canonical benchmarks, including non-smooth geometries. The work provides practical tools for accurate, locking-free simulation of complex shell structures with nonlinear material laws.

Abstract

In this paper we extend the recently introduced mixed Hellan-Herrmann-Johnson (HHJ) method for nonlinear Koiter shells to nonlinear Naghdi shells by means of a hierarchical approach. The additional shearing degrees of freedom are discretized by H(curl)-conforming Nédélec finite elements entailing a shear locking free method. By linearizing the models we obtain in the small strain regime linear Kirchhoff-Love and Reissner-Mindlin shell formulations, which reduce for plates to the originally proposed HHJ and TDNNS method for Kirchhoff-Love and Reissner-Mindlin plates, respectively. By using the Regge interpolation operator we obtain locking-free arbitrary order shell methods. Additionally, the methods can be directly applied to structures with kinks and branched shells. Several numerical examples and experiments are performed validating the excellence performance of the proposed shell elements.

Figures (19)

• Figure 1: Elements with normal, edge tangential, and co-normal vectors.
• Figure 2: Dihedral angle $\sphericalangle(\nu_L,\nu_R)$ between two elements.
• Figure 3: Hierarchical approach for director $\tilde{\nu}\circ\phi$. ${\nu}\circ\phi$ and ${\gamma}\circ\phi$ are the normal and shearing vectors on the deformed configuration.
• Figure 4: Angle computation. (a): With and without averaged normal vector. (b): With averaged normal vector and co-normal vectors. (c): With averaged normal vector from previous step projected to the plane perpendicular to ${\tau}\circ\phi$.
• Figure 5: Left: Kinked shell. Right: Branched shell.

Theorems & Definitions (13)

• Lemma 3.1
• proof
• Lemma 3.2
• proof
• Lemma 4.1
• proof
• Lemma 5.1
• proof
• Lemma 5.2
• proof