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

Yu Zhang,Yidu Yang,Xuqing Zhang

doi:10.3390/math8081252

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.

Highlights

The linear elasticity discusses how solid objects deform and become internally stressed under prescribed loading conditions, and is widely used in structural analysis and engineering design.It has been well-known that using the finite element methods for the elasticity equations/eigenvalue problems, the displacement field can be determined numerically
In order to overcome the phenomenon of locking, various numerical methods for the linear elasticity equations have been developed
Based on the nonconforming CR element Hansbo [6] proposed a discontinuous Galerkin method, the discontinuous Galerkin method is closely related to the classical nonconforming CR element, which is obtained when one of the stabilizing parameters tends to infinity

Summary

Introduction

The linear elasticity discusses how solid objects deform and become internally stressed under prescribed loading conditions, and is widely used in structural analysis and engineering design. Mathematics 2020, 8, 1252 years, mixed nonconforming finite element methods seem to be much more attractive (see [14,15,16,17]), among them, [14,15] studied the linear elasticity equations, [16,17] discussed the elastic eigenvalue problems. We apply the nonconforming CR element to the planar linear elastic eigenvalue problem with the pure displacement and the pure traction boundary conditions, and obtain the guaranteed lower bounds for the eigenvalues. We further develop the work of [44] to obtain the guaranteed lower bounds for eigenvalues by using the nonconforming CR element, and prove that is locking-free (see Theorem 1 for details). Numerical experiments show that using the linear conforming finite element to solve the planar linear elastic eigenvalue problem with the pure traction boundary is locking-free, this is an interesting phenomenon. The bold letters represent vector-valued operators, functions and associated spaces

The Pure Displacement Problem

The Pure Traction Problem

Numerical Experiments

Conclusions