A simple modification to mitigate locking in conforming FEM for nearly incompressible elasticity

Abstract

Due to the divergence-instability, the accuracy of low-order conforming finite element methods (FEMs) for nearly incompressible elasticity equations deteriorates as the Lamé parameter $\hat{I}»\to \infty$, or equivalently as the Poisson ratio $\hat{I}½\to 1/2$. This effect is known as {\itshape locking} or {\itshape non-robustness}. For the piecewise linear case, the error in the ${\mathbf{L}}^2$-norm of the standard Galerkin conforming FEM is bounded by~$C\hat{I}»h^2$, resulting in poor accuracy for practical values of~$h$ if $\hat{I}»$ is sufficiently large. In this short paper, we show that the locking phenomenon can be reduced by replacing $\hat{I}»$ with~$\hat{I}{»}_h=\hat{I}»\hat{I}¼/(\hat{I}¼+\hat{I}»h/L)<\hat{I}»$ in the stiffness matrix, where $\hat{I}¼$ is the second Lamé parameter and $L$ is the diameter of the body $\hat{I}\text{c◯}$. We prove that with this modification, the error in the ${\mathbf{L}}^2$-norm is bounded by $Ch$ for a constant $C$ that does not depend on $\hat{I}»$. Numerical experiments confirm this convergence behaviour and show that, for practical meshes, our method is more accurate than the standard method if $\hat{I}»$ is larger than about $\hat{I}¼L/h$. Our analysis also shows that the error in the ${\mathbf{H}}^1$-norm is bounded by $C\hat{I}{»}_h^{1/2}h$, which improves the $C\hat{I}{»}^{1/2}h$ estimate for the case of conforming FEM.