The finite element and element-free methods for high gradient problems

A general numerical procedure for determining the stress singularity at the vertex of a two material wedge is presented in the first part of this thesis. Using an eigenfunction expansion technique and assuming the asymptotic displacement field or stress potential to be of the form r;In the methods, quadratic and quartic eigenvalue problems are obtained, respectively. The quartic eigenvalue equation ;In the second part of this thesis, an element free Galerkin (EFG) method which is based on the moving least square interpolant (MLS) is discussed. In this method, only a mesh of nodes and a boundary description is needed to develop the Galerkin equations. The interpolants are polynomials which are fit to the nodal values by a weighted least square procedure. A finite element mesh is totally unnecessary in this method.;Some fundamental studies of the EFG method in two dimensional problems of heat conduction and Mindlin plate are presented. The application of EFG to problems of arbitrary fatigue crack growth is also described. The extension of the method to this class of problems involves primarily the development of strategies for moving the denser array of nodes which are needed around the crack tip. An effective strategy which requires approximately 20 nodes about the crack tip and the birth of nodes behind the crack tip to provide high accuracy in stress intensity factors has been developed.