Title:         Locking-Free Finite Elements
               for the Reissner-Mindlin Plate

Authors:       Richard S. Falk and Tong Tu

Source:        Mathematics of Computation, Vol 69 (2000), 911-928

Status:        Published

Abstract:      Two new families of Reissner-Mindlin triangular
               finite elements are analyzed.  One family, generalizing an
               element proposed by Zienkiewicz and Lefebvre, approximates
               (for k >= 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 <= s <= k+1, the
               L^2 errors in the derivatives of the transverse displacement
               are bounded by C h^s and the L^2 errors in the rotation and its
               derivatives are bounded by C h^s min(1, h t^{-1}) and
               C h^{s-1} min(1, h t^{-1}), respectively, for the first family
               and by C h^s and C h^{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
               displacment 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.

Keywords:      Reissner-Mindlin plate, finite element, locking-free

Subj. Class:   65N30, 73K10, 73K25

URL:           http://www.math.rutgers.edu/~falk/locking.pdf