A hierarchic optimisation approach is presented for relieving inaccuracies in conforming shell elements arising from locking phenomena. This approach introduces two sets of strain modes: (i) objective strain modes, defined in the physical coordinate system, and (ii) corrective strain modes, representing conforming strains enhanced with hierarchic strain modes. This leads to two alternative families of element, objective and corrective, both arising from minimising the difference between objective and corrective strains. Importantly, the proposed approach not only alleviates shear and membrane locking, but it also addresses locking arising from element distortion. The application of the proposed optimisation approach is demonstrated for a 9-noded quadrilateral Lagrangian shell element, where the membrane, bending and transverse shear strains are separately optimised, all within a local co-rotational framework that extends the element application to geometric nonlinear analysis. Several numerical examples, including cases with geometric and material nonlinearity, are finally presented to illustrate the effectiveness of the optimised 9-noded shell element in relieving the various sources of locking.

中文翻译：

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一种无锁壳有限元的分层优化方法

提出了一种分层优化方法，以消除因锁定现象引起的一致壳单元的不准确性。这种方法引入了两组应变模式：(i) 在物理坐标系中定义的客观应变模式，以及 (ii) 校正应变模式，表示通过分层应变模式增强的符合应变。这导致了两个可选的元素系列，客观的和修正的，两者都是由于最小化客观应变和修正应变之间的差异而产生的。重要的是，所提出的方法不仅减轻了剪切和膜锁定，而且还解决了由单元变形引起的锁定问题。对 9 节点四边形拉格朗日壳单元演示了所提出的优化方法的应用，其中膜，弯曲和横向剪切应变分别优化，所有这些都在一个局部共旋转框架内，将元素应用扩展到几何非线性分析。最后给出了几个数值例子，包括几何和材料非线性的情况，以说明优化的 9 节点壳单元在解除各种锁定源方面的有效性。