Automatic adaptive refinement for plate bending problems using Reissner-Mindlin plate bending elements

The inßuence of the presence of singular points and boundary layers associated with the edge e¤ects in a ReissnerÐMindlin (RM) plate in the design of an optimal mesh for a Þnite element solution is studied, and methods for controlling the discretization error of the solution are suggested. An e¤ective adaptive reÞnement strategy for the solution of plate bending problems based on the RM plate bending model is developed. This two-stage adaptive strategy is designed to control both the total and the shear error norms of a plate in which both singular points and boundary layers are present. A series of three di¤erent order assumed strain RM plate bending elements has been used in the adaptive reÞnement procedure. The locations of optimal sampling points and the e¤ect of element shape distortions on the theoretical convergence rate of these elements are given and discussed. Numerical experiments show that the suggested reÞnement procedure is e¤ective and that optimally reÞned meshes can be generated. It is also found that all the plate bending elements used can attain their full convergence rates regardless of the presence of singular points and boundary layers inside the problem domain. Boundary layer e¤ects are well captured in all the examples tested and the use of a second stage of reÞnement to control the shear error is justiÞed. In addition, tests on the ZienkiewiczÐZhu error estimator show that their performances are satisfactory. Finally, tests of the relative e¤ectiveness of the plate bending elements used have also been made and it is found that while the higher order cubic element is the most accurate element tested, the quadratic element tested is the most e¦cient one in terms of CPU time used.