A new approach in meshless methods has been introduced for stress assessment around a crack in two-dimensional elastic solids. This method with the name VDLSM (Voronoi Based Discrete Least Squares Meshless) is a pure meshless method which does not implement nodal mesh for trial and test functions. Rather, it uses a collection of the scattered nodal points and implements the discrete least squares approach to discretize the strong form of the governing differential equations on the domain of interest. This can reduce considerably the pre-processing cost of the analysis. Meshless methods generally are faced with some difficulty to accommodate the stress analysis in the vicinity of sharply concave surfaces such as cracks. Some techniques have been used to fix that problem such as visibility, transparency and diffraction, but these methods require some additional back corrective analyses for those parts of the domain located near the crack, which lengthen the time consumed for the solution and, moreover, do not provide the desired accuracy for the unknowns in these regions. VDLSM as a new, straightforward and easy applicable method has been suggested here for overcoming such deficiency using the algorithm of the Voronoi tessellation for constructing the Moving Least Squares (MLS) shape functions.

This study proposes a new formulation of singular boundary method (SBM) to solve the 2D potential problems, while retaining its original merits being free of integration and mesh, easy-to-program, accurate and mathematically simple without the requirement of a fictitious boundary as in the method of fundamental solutions (MFS). The key idea of the SBM is to introduce the concept of the origin intensity factor to isolate the singularity of fundamental solution so that the source points can be placed directly on the physical boundary. This paper presents a new approach to derive the analytical solution of the origin intensity factor based on the proposed subtracting and adding-back techniques. And the troublesome sample nodes in the ordinary SBM are avoided and the sample solution is also not necessary for the Neumann boundary condition. Three benchmark problems are tested to demonstrate the feasibility and accuracy of the new formulation through detailed comparisons with the boundary element method (BEM), MFS, regularized meshless method (RMM) and boundary distributed source (BDS) method.