Locking-free finite elements for the Reissner-Mindlin plate

Two new families of Reissner-Mindlin triangular finite elements are analyzed. One family, generalizing an element proposed by Zienkiewicz and Lefebvre, approximates (for $k\ge 1)$ the transverse displacement by continuous piecewise polynomials of degree $k+1$, the rotation by continuous piecewise polynomials of degree $k+1$ plus bubble functions of degree $k+3$, and projects the shear stress into the space of discontinuous piecewise polynomials of degree $k$. The second family is similar to the first, but uses degree $k$ rather than degree $k+1$ continuous piecewise polynomials to approximate the rotation. We prove that for $2\le s\le k+1$, the $L^2$ errors in the derivatives of the transverse displacement are bounded by $Ch^s$ and the $L^2$ errors in the rotation and its derivatives are bounded by $Ch^s\min(1,ht^{-1})$ and $Ch^{s-1}\min(1,ht^{-1})$, respectively, for the first family, and by $Ch^s$ and $Ch^{s-1}$, respectively, for the second family (with $C$ independent of the mesh size $h$ and plate thickness $t)$. These estimates are of optimal order for the second family, and so it is locking-free. For the first family, while the estimates for the derivatives of the transverse displacement are of optimal order, there is a deterioration of order $h$ in the approximation of the rotation and its derivatives for $t$ small, demonstrating locking of order $h^{-1}$. Numerical experiments using the lowest order elements of each family are presented to show their performance and the sharpness of the estimates. Additional experiments show the negative effects of eliminating the projection of the shear stress.