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

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. Copyright © 2004 John Wiley & Sons, Ltd.