This paper uses a locking-free nonconforming Crouzeix–Raviart finite element to solve the planar linear elastic eigenvalue problem with homogeneous pure displacement boundary condition. Making full use of the Poincaré inequality, we obtain the guaranteed lower bounds for eigenvalues. Besides, we also use the nonconforming Crouzeix–Raviart finite element to the planar linear elastic eigenvalue problem with the pure traction boundary condition, and obtain the guaranteed lower eigenvalue bounds. Finally, numerical experiments with MATLAB program are reported.

中文翻译：

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通过使用不合格的Crouzeix–Raviart有限元，保证弹性特征值的下界

本文使用无锁定非协调Crouzeix–Raviart有限元解决了具有均质纯位移边界条件的平面线性弹性特征值问题。充分利用庞加莱不等式，我们可以获得特征值的有保证的下界。此外，对于纯牵引边界条件的平面线性弹性特征值问题，我们还使用了非协调Crouzeix–Raviart有限元，并获得了有保证的特征值下界。最后，报道了用MATLAB程序进行的数值实验。