The fractal finite element method for unbounded problems

Abstract

The fractal finite element method, previously developed for stress intensity factor calculation for crack problems in fracture mechanics, is extended to analyse some unbounded problems in half space. The concepts of geometrical similarity and two-level finite element mesh are applied to generate an infinite number of self-similar layers in the far field with a similarity ratio bigger than one; that is, one layer is bigger than the next in size but of the same shape. Only conventional finite elements are used and no new elements are generated. The global interpolating functions in the form of a truncated infinite series are employed to transform the infinite number of nodal variables to a small number of unknown coefficients associated with the global interpolating functions. Taking the advantage of geometrical similarity, transformation for one layer is enough because the relevant entries of the transformed matrix after assembling all layers are infinite geometric series of the similarity ratio and can be summed analytically. Accurate nodal displacements are obtained as shown in the numerical examples.