Fractal finite element method for singular problems

A semi-analytical method is proposed for solving singular problems, in particular, determination of stress intensity factors (SIF) of elastic crack problems. In this method, the singularity is eliminated from the computational domain by the fractal finite element method (FFEM). The fractal geometry concept and two level finite element method (2LFEM) are employed to automatically generate an infinitesimal mesh and transform these large number of degrees of freedom (DOF) around the singular regions, such as crack tip, to a small set of generalized coordinates. The dimensional independence of stiffness for similar finite elements enables us to evaluate the stiffness matrices of one layer of mesh only with the others thereby obtained readily. Not similar to the situation in conventional finite element where a considerable amount of unknowns have to be dealt with, what remains here are only a small set of generalized coordinates. In this paper, the formulation of the method will be discussed, a simple hand calculation example will be provided to illustrate the principle and the effectiveness of the method. A series of applications of this method will then be presented.