One of the major challenges in computational mechanics is the development of numerical methods for resolving the structure of functions with high gradients in complicated geometries. In problems such as material failure due to localization and crack propagation, or shocks in fluid dynamics, the regions of high gradients are of several orders of magnitude smaller than the size of the domain of interest. Therefore, if a uniform finite element mesh is used, a tremendous amount of elements would be needed to accurately resolve the structure of high gradients. The purpose of this research is to improve the quality of finite element calculations in the regions of high gradients. This is accomplished by superimposing a patch of higher order hierarchical elements on the portion of the original finite element mesh where resolution is required. Continuity of the displacement field is maintained by imposing homogenous boundaries between the original and superimposed fields. The superimposed and the original mesh topology. The research to be carried out as part of this project will consider the following aspects: 1. The development and implementation of the finite element mesh superposition method. 2. Automation of the superposition method. High resolution of the fields will be obtained by controlling the location of the superimposed field, its subdivision and the polynomial order of the superimposed elements. The process of selection of these variables will be steered by a posteriori information. Pointwise error and the superimposed mesh will be designed to make a nearly optimal use of posterior information.
