B-spline wavelet finite element method to the elastostatic crack problems is proposed. B-spline wavelet finite element method (Tanaka et. al., 2006) is a new technique for solving solid/structural mechanics problems. B-spline scaling function and wavelets are used as the basis functions in the Galerkin formulation. These wavelet basis functions have the so-called multiresolution properties. The solutions can be refined in regions of high gradients such as stresses or strains by superposing different lengths scale wavelets. Adaptive strategies using the multiresolution properties are developed for the elastostatic solid/structural problems (Tanaka et. al., 2007). On the other hand, there are difficulties in dealing with discontinuous displacements for the analysis of crack problems because the continuities of the displacements are always guaranteed the B-spline wavelet finite element method. Then, we propose a new technique. The displacements are enriched by discontinuous functions and the near tip asymptotic solutions through a partition of unity method (Melenk, 1996; Babuska, 1997). In this paper, emphasis is placed on linear B-spline scaling function as the basis functions in Galerkin formulation. Mathematical formulation and numerical implementations for the crack problems are presented. Some numerical examples for the two dimensional elastostatic crack problems are shown.
The objective of this study is to solve the solid/structural problems using B-spline wavelet finite element method. It has been known that the use of wavelet is a new technique for solving partial differential equations, as well as signal processing, image processing and function approximation (Charles, 1992; Williams, 1994). A signal or a function is interpolated by scaling function and wavelets in the wavelet analysis. Wavelet basis functions can divide a given signal or function into different length scale component. These features of the wavelet basis functions are the so-called multiresolution properties. A various kinds scaling function and wavelets have been proposed for differentpurposes.