L. Chen

It is well known that the lower bound to exact solutions in linear fracture problems can be easily obtained by the displacement compatible finite element method (FEM) together with the singular crack tip elements. It is, however, much more difficult to obtain the upper bound solutions for these problems. This paper aims to formulate a novel singular node‐based smoothed finite element method (NS‐FEM) to obtain the upper bound solutions for fracture problems. In the present singular NS‐FEM, the calculation of the system stiffness matrix is performed using the strain smoothing technique over the smoothing domains (SDs) associated with nodes, which leads to the line integrations using only the shape function values along the boundaries of the SDs. A five‐node singular crack tip element is used within the framework of NS‐FEM to construct singular shape functions via direct point interpolation with proper order of fractional basis. The mix‐mode stress intensity factors are evaluated using the domain forms of the interaction integrals. The upper bound solutions of the present singular NS‐FEM are demonstrated via benchmark examples for a wide range of material combinations and boundary conditions. Copyright © 2010 John Wiley & Sons, Ltd.

A generalized gradient smoothing technique and a smoothed bilinear form of Galerkin weak form have been recently proposed by Liu et al. to create a wide class of efficient smoothed point interpolation methods (PIMs) using the background mesh of triangular cells. In these methods, displacement fields are constructed by polynomial or radial basis shape functions and strains are smoothed over the smoothed domain associated with the nodes or the edges of the triangular cells. This paper summarizes and assesses bound property, convergence rate and computational efficiency for these methods. It is found that: (1) the incorporation of the PIMs with the node‐based strain smoothing operation allows us to obtain an upper bound to the exact solution in the strain energy; (2) the incorporation of the PIMs with the edge‐based strain smoothing operation using triangular background mesh can produce a solution of ‘ultra‐accuracy’ and ‘super‐convergence’; (3) the edge‐based strain smoothing operation together with the linear interpolation can provide better computational efficiency compared with other smoothed PIMs and the finite element method when the same triangular mesh is used. These conclusions have been examined and confirmed by intensive examples. Copyright © 2009 John Wiley & Sons, Ltd.