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

Abstract

SUMMARYThe 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.

Two-dimensional fractal-like finite element method for thermoelastic crack analysis

“…More information about the FFEM can be found in previous work (Tsang et al, 2004). The overall crack problem is divided into near field and far field regions as shown in Fig.…”

“…The William's eigenfunction can be found analytically for most two-dimensional crack problems. In the case that an analytical solution cannot be found, a numerical method has been developed to evaluate the eigenfunctions numerically (Tsang et al, 2004). In this study, however, the analytical approach was used.…”

“…Fractal-like finite element method (FFEM) (Leung and Tsang, 2000;Tsang et al, 2003Tsang et al, , 2004Tsang and Oyadiji, 2005a) is one of the few finite element methods which can determine SIF directly. The FFEM has been proved to be very efficient and accurate in two-dimensional static and dynamic crack problems.…”

Super singular element method for two-dimensional crack analysis

“…The SSEM is an improved version of the fractallike finite-element method (Leung et al 2000;Tsang et al 2004Tsang et al , 2007. The SSEM is designed to be easy to use for engineering purposes such that crack analysis can be done by a normal stress analyst.…”

“…The super singular element method (SSEM) is a semi-analytical method, which combines the efficiency and accuracy of analytical solutions and the agility of standard finite-element method. The SSEM is an improved version of the fractallike finite-element method (Leung et al 2000;Tsang et al 2004Tsang et al , 2007. The SSEM is designed to be easy to use for engineering purposes such that crack analysis can be done by a normal stress analyst.…”