Enhanced stress projections for error estimation and a singular floating node element for static and dynamic fracture

This thesis presents two new developments: (1) an enhanced stress recovery method for error estimation, and (2) a new singular element for static and dynamic crack propagation. Applications of these techniques to various problems are presented.;Error estimation is an essential task for adaptive analyses using the finite element method. Improvements in the recovery of derivative quantities, e.g. stress and strain, allow for a better approximation of the exact derivatives. It is shown that adding the square of the boundary condition and the equilibrium residual to both local and global stress projection substantially improves the accuracy, but not the rate of convergence. In addition, a new conjoint polynomial for interpolating the local patch stresses is presented which significantly improves the local projection scheme. Several examples are presented which show the utility of these enhancements.;A sizing function which uses the estimated error for setting element size for adaptive remeshing is also presented. This sizing function is based on theoretical convergence rates and size gradient controls. The paving meshing algorithm is used to remesh the domain and adaptive iterations are performed.;A new singular floating node element has been devised for numerical analysis of static fatigue crack propagation as well as rapid crack propagation in dynamic fracture. A fifth node is added to the edge of the standard 4-node quadrilateral element. This floating node remains at the crack tip as the crack propagates, providing a convenient and accurate method of unzipping the mesh during crack propagation. A variable order singularity has been added to the element. The shape function associated with the floating node can be specified with a great deal of freedom, and several choices for this function were explored. The performance of this element is demonstrated in a static crack analysis, in analysis of dynamic loading with a stationary crack, and in dynamic crack propagation.