R-H mesh improvement algorithms for the finite element method

New algorithms based on an r-h approach for mesh improvement of displacement finite element discretizations are developed. The algorithms are applied to one and two dimensional elasticity problems. The scheme can be used as part of an efficient preprocessor for general purpose finite element codes to improve the solution. Different approaches for error estimators for finite element meshes are first reviewed. Measures of the convergence of adaptive mesh improvement processes are proposed and compared. A new mixed algorithm combining r improvements (mesh modification) and h improvements (mesh refinements) is proposed, and applied to a rod example in a preliminary parametric study. The algorithm is extended to plane problems, where different strategies are developed for the calculation and minimization of error estimators in two dimensional spaces. The coupling of the two improvement schemes is done with the h refinement performed in a global and nested form. The introduction of an interweaving algorithm where the adaptive cycles resort to iterative techniques for the solution of the systems of equations allows for the use of intermediary results to accelerate convergence. Different norms are studied for the error estimators, and the effect of geometric constraints on the modification process is analyzed. The method is then modified for treating a different class of engineering applications. This group of problems deals with verification processes, where an approximation to the solution is known a priori . Error measures and improvement procedures are developed and applied to a number of examples. The examination of a series of applications to adaptive and verification example problems indicates that the use of r-h improvement techniques provides a viable approach to generate efficient grids, using simple data structures and obtaining reasonable computational performance and appropriate discrete models