Finite elements with Allman's rotations provide good computational efficiency for explicit codes exhibiting less locking than linear elements and lower computational cost than quadratic finite elements. One way to further raise their efficiency is to increase the feasible time step or increase the accuracy of the lowest eigenfrequencies via reciprocal mass matrices. This article presents a formulation for variationally scaled reciprocal mass matrices and an efficient estimator for the feasible time step for finite elements with Allman's rotations. These developments take special care of two core features of such elements: existence of spurious zero‐energy rotation modes implying the incompleteness of the ansatz spaces, and the presence of mixed‐dimensional degrees of freedom. The former feature excludes construction of dual bases used in the standard variational derivation of reciprocal mass matrices. The latter feature destroys the efficiency of the existing nodal‐based time step estimators stemming from the Gershgorin's eigenvalue bound. Finally, the developments are tested for standard benchmarks and triangular, quadrilateral, and tetrahedral finite elements with Allman's rotations.

中文翻译：

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具有Allman旋转的有限元的倒数质量矩阵和可行的时间步估计

具有Allman旋转的有限元为显式代码提供了良好的计算效率，与线性元素相比，其显示出的锁定更少，而与二次有限元相比，其计算成本更低。进一步提高其效率的一种方法是通过倒数质量矩阵来增加可行的时间步长或提高最低本征频率的精度。本文提出了变分比例互易质量矩阵的公式，并给出了具有Allman旋转的有限元可行时间步长的有效估计。这些发展特别注意了此类元素的两个核心特征：虚假的零能量旋转模式的存在暗示了ansatz空间的不完整性，以及存在混合维度的自由度。前一个特征不包括用于对等质量矩阵的标准变分推导的对偶碱基的构造。后一特征破坏了现有的基于Gershgorin特征值界限的基于节点的时间步长估计器的效率。最后，使用Allman旋转对开发的标准基准以及三角形，四边形和四面体有限元进行测试。