In this paper, the finite element discretization of non-classical continuum models with micro-inertia is analysed. The focus is on micro-inertia extensions of the one-dimensional rod model, the beam bending theories of Euler–Bernoulli and Rayleigh, and the two-dimensional membrane model. The performance of a variety of mass matrices is assessed by comparing the natural frequencies and their modes with those of the associated discrete systems, and it is demonstrated that the use of higher-order mass matrices reduces errors and improves convergence rates. Furthermore, finite element sizes larger than the corresponding physical length scale are shown to be sufficient to capture the natural frequencies, thus facilitating numerical models that are not only reliable but also computationally efficient.

本文分析了具有微惯性的非经典连续体模型的有限元离散化。重点是一维杆模型的微惯性扩展、欧拉-伯努利和瑞利的梁弯曲理论以及二维膜模型。通过将自然频率及其模式与相关离散系统的自然频率及其模式进行比较，评估了各种质量矩阵的性能，并证明使用高阶质量矩阵可以减少错误并提高收敛速度。此外，大于相应物理长度尺度的有限元尺寸被证明足以捕获自然频率，从而促进不仅可靠而且计算效率高的数值模型。