Guaranteed Lower Bounds for the Elastic Eigenvalues by Using the Nonconforming Crouzeix–Raviart Finite Element

Abstract

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.

Keywords

• elastic eigenvalue problem
• lower eigenvalue bounds
• the Poincaré inequality
• nonconforming Crouzeix–Raviart finite element