Remarks on Quadrilateral Reissner-Mindlin Plate Elements

Over the last two decades, there has been an extensive effort to devise and analyze £nite elements schemes for the approximation of the Reissner–Mindlin plate equations which avoid lock ng, numerical overstiffness resulting in a loss of accuracy when the plate is thin. There are now many triangular and rectangular £nite elements, for which a mathematical analysis exists to certify them as free of locking. Generally speaking, the analysis for rectangular elements extends to the case of parallograms, which are de£ned by af£ne mappings of rectangles. However, for more general convex quadrilaterals, de£ned by bilinear mappings of rectangles, the analysis is more complicated. Recent results by the authors on the approximation properties of quadrilateral £nite elements shed some light on the problems encountered. In particular, they show that for some £nite element methods for the approximation of the Reissner-Mindlin plate, the obvious generalization of rectangular elements to general quadrilateral meshes produce methods which lose accuracy. In this paper, we present an overview of this situation. Douglas N. Arnold, Daniele Bof£, Richard S. Falk