In this paper, we initiate the use of spectral analysis for assessing locking phenomena in finite element formulations. We propose to ``measure'' locking by comparing the difference between eigenvalue and mode error curves computed on coarse meshes with ``asymptotic'' error curves computed on ``overkill'' meshes, both plotted with respect to the normalized mode number. To demonstrate the intimate relation between membrane locking and spectral accuracy, we focus on the example of a circular ring discretized with isogeometric curved Euler-Bernoulli beam elements. We show that the transverse-displacement-dominating modes are locking-prone, while the circumferential-displacement-dominating modes are naturally locking-free. We use eigenvalue and mode errors to assess five isogeometric finite element formulations in terms of their locking-related efficiency: the displacement-based formulation with full and reduced integration and three locking-free formulations based on the B-bar, discrete strain gap and Hellinger-Reissner methods. Our study shows that spectral analysis uncovers locking-related effects across the spectrum of eigenvalues and eigenmodes, rigorously characterizing membrane locking in the displacement-based formulation and unlocking in the locking-free formulations.

中文翻译：

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利用光谱分析阐明弯曲 Euler-Bernoulli 梁的等几何有限元公式中的膜锁定和解锁

在本文中，我们开始使用谱分析来评估有限元公式中的锁定现象。我们建议通过比较在粗网格上计算的特征值和模式误差曲线与在“过度杀伤”网格上计算的“渐近”误差曲线之间的差异来“测量”锁定，两者都根据归一化模式数绘制。为了证明膜锁定和光谱精度之间的密切关系，我们将重点放在使用等几何弯曲欧拉-伯努利梁单元离散化的圆环的示例上。我们表明横向位移主导模式易于锁定，而周向位移主导模式自然无锁定。我们使用特征值和模式误差来评估五个等几何有限元公式的锁定相关效率：基于位移的完全和减少积分公式和三个基于 B 杆、离散应变间隙和海灵格的无锁定公式-赖斯纳方法。我们的研究表明，谱分析揭示了特征值和特征模式谱中的锁定相关效应，严格表征了基于位移的公式中的膜锁定和无锁定公式中的解锁。