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Mathematics of Computation
Mathematics of Computation
ISSN 1088-6842(online) ISSN 0025-5718(print)

 

Locking-free finite elements
for the Reissner-Mindlin plate


Authors: Richard S. Falk and Tong Tu
Journal: Math. Comp. 69 (2000), 911-928
MSC (1991): Primary 65N30, 73K10, 73K25
Published electronically: August 20, 1999
MathSciNet review: 1665950
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Abstract: Two new families of Reissner-Mindlin triangular finite elements are analyzed. One family, generalizing an element proposed by Zienkiewicz and Lefebvre, approximates (for $k\ge 1)$ 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.


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Additional Information

Richard S. Falk
Affiliation: Department of Mathematics, Rutgers University, Piscataway, New Jersey 08854
Email: falk@math.rutgers.edu

Tong Tu
Affiliation: Bloomberg Princeton Index Group, 100 Business Park Drive, Skillman, New Jersey 08858
Email: tongtu@bloomberg.net

DOI: http://dx.doi.org/10.1090/S0025-5718-99-01165-5
PII: S 0025-5718(99)01165-5
Keywords: Reissner-Mindlin plate, finite element, locking-free
Received by editor(s): August 14, 1998
Published electronically: August 20, 1999
Additional Notes: The first author was supported by NSF grant DMS-9704556
Article copyright: © Copyright 2000 American Mathematical Society