Applications of numerical eigenfunctions in the fractal-like finite element method

Abstract

The fractal-like finite element method is an accurate and efficient method to determine the stress intensity factors (SIF) around crack tips. The use of a self-similar mesh together with the William's eigenfunctions reduce the number of unknowns significantly. The SIFs are the primary unknowns and no post-processing is required. In all previous studies, we used the analytic eigenfunction expression to perform the global transformation. However, the eigenfunction cannot be found analytically in general crack problems. Two-dimensional axisymmetrical cracks are considered here. The resulting static equilibrium equations in local co-ordinates are non-homogeneous ordinary differential equations, for which the analytic eigenfunction cannot be found completely. We use a finite difference method to determine all the eigenfunctions needed numerically. Our evaluated SIF values show very close agreement with published results.