Coupling Of Fractal Finite Element Method And Meshless Method For Fracture Mechanics Problems

The fractal finite element method(FFEM),which separates the computational domain into a singular region near the crack tip and a regular region far from the crack tip,is a new numerical method to deal with linear elastic fracture mechanics problems.In this method,both the regular and the singular regions are discreted with conventional finite element and the Williams eigenfunction expansion is introduced in the singular region.Within the singular region an infinite number of elements are generated by the self-similarity of elements and the fractal process to capture the singular behavior at the crack tip.This method does not need to introduce special enriched basis functions,and the stress intensity factor can be obtained without post-processing.However,this method also encounters problems such as difficulty in meshing and mesh distortion in the regular region.Having the advantages of convenient preprocessing and easy formation of high-order approximations,the meshless method which only needs to arrange nodes without meshing has emerged in recent years.Interpolating element-free Galerkin method(IEFG),as a type of meshless method,overcomes the shortcoming of the shape function constructed by most meshless methods that do not have interpolation.This method uses the interpolating moving least-squares to establish shape functions with interpolation.Moreover,the shape function formula is simpler than Lancaster's formula,which can improve the calculation efficiency.In order to combine the best features of FFEM and IEFG,this paper proposes a coupling FFEM and IEFG method to deal with fracture mechanics.In this coupling method,the IEFG is used to discretize the calculation domain in the regular region far from the crack tip,and the FFEM is used to discretize the calculation domain in the singular region near the crack tip.At the same time,the coupling method makes use of the interpolation characteristics of the shape functions of the FFEM and IEFG,by setting the points on the interface of the two methods not only as fractal finite element nodes,but also as meshless nodes so as to ensure the continuity of the two methods on the interface.Based on the FFEM and IEFG,the driect coupling method of FFEM and IEFG is proposed to solve linear elastic fracture mechanics problems in this paper.In addition,the corresponding Matlab program has been developed.At last,a lot of numerical examples are presented to verify the effectiveness and accuracy of the proposed method for linear elastic fracture mechanics problems.