埃尔米特梁单元的分块对角与高阶质量矩阵

埃尔米特梁单元常用的集中质量矩阵，是由挠度自由度对应的一致质量矩阵元素通过行求和或节点积分构造。然而，数值结果表明该集中质量矩阵在求解包含自由端的梁振动问题时，会出现频率精度掉阶现象。本文首先从保障质量矩阵最优收敛性的数值积分精度出发，分别针对三次和五次梁单元，发展了质量矩阵的梯度增强节点积分方案。利用梯度增强节点积分方案，可以得到具有分块对角形式的单元质量矩阵，而其组装的整体质量矩阵除边界节点外仍然呈现对角形式。对于两种单元，其分块对角质量矩阵分别具有4阶最优精度和6阶次优精度。再者，将标准一致质量矩阵和具有同阶精度的梯度增强节点积分质量矩阵进行优化组合，建立了具有超收敛特性的高阶质量矩阵。最后，通过数值算例系统验证了三次和五次单元的分块对角与高阶质量矩阵的频率计算精度。

Block diagonal and higher order mass matrices for Hermite beam elements

The lumped mass matrices of Hermite finite elements for Euler-Bernoulli beams are often constructed by the row sum technique via neglecting the rotational entries, or by the nodal quadrature.However, such lumped mass matrices of Hermite beam elements exhibit a sudden drop of accuracy in frequency calculation in case of free boundary conditions.In this study, based upon the numerical integration accuracy requirement, the gradient-enhanced nodal quadrature rules are developed for both cubic and quintic elements.These nodal quadrature rules lead to a block diagonal form of element mass matrices, while the assembled global mass matrix still preserves a desirable diagonal pattern except a few entries associated with the boundary nodes.The block diagonal mass matrices of cubic and quintic elements have an optimal convergence rate of 4 and a sub-optimal convergence rate of 6, respectively.Subsequently, through rationally mixing the consistent mass matrices and the mass matrices generated by the gradient-enhanced nodal quadrature rules with equal accuracy orders, superconvergent higher-order mass matrices are attained for cubic and quintic Hermite beam elements.The accuracy and robustness of the proposed block diagonal and higher-order mass matrices are systematically demonstrated by numerical results.