Use of hierarchical elements in isoparametric finite element procedures

Abstract

This thesis reports on an initial study into how hierarchical elements can best be used to enhance existing isoparametric finite element routines. Hierarchical finite elements have been used to date primarily for self-adaptive solutions, but the architecture of these programs is significantly different than conventional finite element programs. The goal is to develop methods by which hierarchical elements can be easily adapted to the large number of existing conventional programs and provide additional output to assist the analyst. The investigation discussed in this thesis is carried out in one dimension. It is demonstrated that a lower-order formulation with a higher-order hierarchical term added results in a solution equivalent to the complete higher-order solution. A technique is demonstrated by which a correct higher-order solution can be iteratively calculated using the initial lower-order stiffness matrix and without assembly of the higher-order stiffness terms. Also shown is a method for calculating a useful element-level error indicator. The one-dimensional model problem used is of a form which yields non-zero off-diagonal hierarchical stiffness terms; a situation typical in cases of higher dimension. Consequently, this work provides a basis for extension of the concepts into higher dimensions, where the benefit of the hierarchical enhancement will be most significant.