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SUMMARYThe estimation of the parameters ('fictitious densities') which control the convergence and numerical stability of a non-linear Dynamic Relaxation solution is described. The optimal values of these parameters vary during the iterative solution and they are predicted from the Gerschgorin bounds, that is rowsums of the stiffness matrix, which are divided into constant and variable parts for computational convenience. The procedure is illustrated by reference to the analysis of an axially loaded beam on a non-uniform elastic foundation.

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The influence of the presence of singular points and boundary layers associated with the edge effects in a Reissner-Mindlin (RM) plate in the design of an optimal mesh for a finite element solution is studied, and methods for controlling the discretization error of the solution are suggested. An effective adaptive refinement 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 different order assumed strain RM plate bending elements has been used in the adaptive refinement procedure. The locations of optimal sampling points and the effect of element shape distortions on the theoretical convergence rate of these elements are given and discussed. Numerical experiments show that the suggested refinement procedure is effective and that optimally refined 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 effects are well captured in all the examples tested and the use of a second stage of refinement to control the shear error is justified. In addition, tests on the Zienkiewicz-Zhu error estimator show that their performances are satisfactory. Finally, tests of the relative effectiveness 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 efficient one in terms of CPU time used.