This paper presents a coupling technique for integrating the element–free Galerkin method (EFGM) with fractal two-level finite element method (F2LFEM) for analyzing homogeneous, isotropic, and two dimensional linear–elastic cracked structures subjected to mixed–mode (modes I and II) loading conditions. F2LFEM is adopted for discretization of domain close to the crack tip and EFGM is adopted in the rest of the domain. In the transition region interface elements are employed. The shape functions within interface elements which comprises both the element–free Galerkin and the finite element shape functions, satisfies the consistency condition thus ensuring convergence of the proposed method. The proposed method combines the best features of EFGM and F2LFEM, in the sense that no structured mesh or special enriched basis functions are necessary and no post–processing (employing any path independent integrals) is needed to determine fracture parameters such as stress–intensity factors (SIFs) and T–stress. The numerical results show that SIFs and T–stress obtained using the proposed method are in excellent agreement with the reference solutions for the structural and crack geometries considered in this study. Also a parametric study is carried out to examine the effects of the similarity ratio, and the number of transformation terms on the quality of the numerical solutions.
A multi-deformable bodies solution method coupling finite element with meshless method in sheet metal flexible-die forming