New methods are presented for the direct computation of higher-order inverse mass matrices (also called reciprocal mass matrices) that are used for explicit transient finite element analysis. The motivation of this work lies in the need of having appropriate sparse inverse mass matrices, which present the same structure as the consistent mass matrix, preserve the total mass, predict suitable frequency spectrum and dictate sufficiently large critical time step sizes. For an efficient evaluation of the reciprocal mass matrix, the projection matrix should be diagonal. This condition can be satisfied by adopting dual shape functions for the momentum field, generated from the same shape functions used for the displacement field. A theoretically consistent derivation of the inverse mass matrix is based on the three-field Hamilton principle and requires the projection matrix to be evaluated from the integral of these shape functions. Unfortunately, for higher-order FE shape functions and serendipity FE elements, the projection matrix is not positive definitive and can not be employed. Therefore, we study several lumping procedures for higher order reciprocal mass matrices considering their effect on total-mass preserving, frequency spectra and accuracy in explicit transient simulations. The article closes with several numerical examples showing suitability of the direct inverse mass matrix in dynamics.

中文翻译：

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用于异质固体显式瞬态分析的高阶逆质量矩阵

提出了直接计算用于显式瞬态有限元分析的高阶逆质量矩阵（也称为倒数质量矩阵）的新方法。这项工作的动机在于需要拥有适当的稀疏逆质量矩阵，该矩阵呈现与一致质量矩阵相同的结构，保留总质量，预测合适的频谱并指示足够大的临界时间步长。为了有效地评估质量倒数矩阵，投影矩阵应该是对角的。这个条件可以通过采用动量场的双形函数来满足，该双形函数是由用于位移场的相同形函数生成的。逆质量矩阵的理论上一致的推导基于三场哈密顿原理，并且需要根据这些形状函数的积分来评估投影矩阵。不幸的是，对于高阶有限元形状函数和偶然性有限元元素，投影矩阵不是正定的，无法使用。因此，我们研究了高阶倒数质量矩阵的几种集总程序，考虑它们对显式瞬态模拟中总质量保持、频谱和精度的影响。本文最后用几个数值例子来展示直接逆质量矩阵在动力学中的适用性。