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