Purpose

A general strategy is developed for adaptive finite element (FE) analysis of free vibration of elastic membranes based on the element energy projection (EEP) technique.

Design/methodology/approach

By linearizing the free vibration problem of elastic membranes into a series of linear equivalent problems, reliable a posteriori point-wise error estimator is constructed via EEP super-convergent technique. Hierarchical local mesh refinement is incorporated to better deal with tough problems.

Findings

Several classical examples were analyzed, confirming the effectiveness of the EEP-based error estimation and overall adaptive procedure equipped with a local mesh refinement scheme. The computational results show that the adaptively-generated meshes reasonably catch the difficulties inherent in the problems and the procedure yields both eigenvalues with required accuracy and mode functions satisfying user-preset error tolerance in maximum norm.

Originality/value

By reasonable linearization, the linear-problem-based EEP technique is successfully transferred to two-dimensional eigenproblems with local mesh refinement incorporated to effectively and flexibly deal with singularity problems. The corresponding adaptive strategy can produce both eigenvalues with required accuracy and mode functions satisfying user-preset error tolerance in maximum norm and thus can be expected to apply to other types of eigenproblems.

中文翻译：

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基于元能量投影技术的弹性膜自由振动自适应有限元分析

目的

基于元能量投影 (EEP) 技术，开发了一种通用策略，用于弹性膜自由振动的自适应有限元 (FE) 分析。

设计/方法/方法

通过将弹性膜的自由振动问题线性化为一系列线性等价问题，利用EEP超收敛技术构建了可靠的后验点误差估计器。分层局部网格细化被纳入以更好地处理棘手的问题。

发现

分析了几个经典示例，确认了基于 EEP 的误差估计和配备局部网格细化方案的整体自适应程序的有效性。计算结果表明，自适应生成的网格合理地捕捉了问题中固有的困难，并且该过程产生了具有所需精度的特征值和满足用户预设误差最大范数的模式函数。

原创性/价值

通过合理的线性化，将基于线性问题的 EEP 技术成功地转化为二维特征问题，并结合局部网格细化，有效灵活地处理奇异性问题。相应的自适应策略可以产生具有所需精度的特征值和在最大范数下满足用户预设容错的模式函数，因此可以预期应用于其他类型的特征问题。