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 methods for Kirchhoff–Love and Reissner–Mindlin plates, respectively. By interpolating the membrane strains into the so-called Regge finite element space 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 excellent performance of the proposed shell elements.

中文翻译：

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线性和非线性壳的 Hellan-Herrmann-Johnson 和 TDNNS 方法


在本文中，我们通过分层方法将最近引入的非线性 Koiter 壳的 Hellan-Herrmann-Johnson （HHJ） 混合方法扩展到非线性 Naghdi 壳。额外的剪切自由度由符合 H（curl） 的 Nédélec 有限元离散，该有限元需要无剪切锁定方法。通过线性化我们在小应变状态线性 Kirchhoff-Love 和 Reissner-Mindlin 壳公式中获得的模型，这些公式将板简化为最初提出的 Kirchhoff-Love 和 Reissner-Mindlin 板的 HHJ 和 TDNNS 方法。通过将膜应变插入所谓的 Regge 有限元空间，我们获得了无锁定的任意阶壳方法。此外，这些方法可以直接应用于具有扭结和支链壳的结构。进行了几个数值示例和实验，验证了所提出的壳单元的优异性能。