Coupled Fractal Finite Element And Boundary Element Method For Analyses Of Fracture Problems

The stress intensity factors(SIFs)are important parameters in fracture mechanics,which have a wide range of applications in engineering fracture analysis.The fractal finite element method(FFEM)is a new numerical method for solving linear elastic fracture mechanics problems.The FFEM is based on the concept of fractal geometry,which generates an infinite number of self-similar elements around the crack tip.The nodal displacements in the singular region are transformed to a set of generalized coordinates using Williams eigenfunction expansion for the displacement fields near the crack tip.This significantly reduces the number of unknown coefficients,and allows the SIFs to be directly obtained based on these generalized coordinates.The FFEM can eliminate the singular behavior in the calculation around the crack and reduce the computational time and the memory requirement for fracture analysis.In addition,the boundary element method(BEM)is an accurate and efficient numerical method.The basic idea of BEM is to transform the differential equation on the solution domain into a boundary integral equation,so that the analysis of the problem is then transferred to the boundary of the solution domain,and then interpolation is performed on the boundary to obtain a discrete equation.The physical quantities of points inside the domain can be directly calculated using analytical formulas based on the solutions on the boundary,thus the computational accuracy is greatly improving.The BEM discretizes only the boundaries of the problem domain,without requiring the partitioning of the grid within the domain.This reduces the spatial dimensionality of the problem domain,greatly decreasing the amount of input data and computational work required.Additionally,quadratic discontinuous elements are employed to discretize the boundary integral equation so that the corner problem that affects the accuracy of the boundary element analysis can be well handled.In this paper,a novel numerical method is presented for fracture analyses by coupling FFEM with BEM,which can not only make full use of the advantages of FFEM but also the advantages of BEM.In addition,the corresponding computational programs are developed.Ultimately,a simple,accurate and efficient computational theory and method is constructed for the simulation of fracture problems.The coupling technique separates the cracked body into a regular and a singular region,with the latter enclosing the crack tip.The regular and the singular regions are modeled by the BEM and the FFEM,respectively.The boundary element part in the whole computational model is defined as a super element of the finite element method,and its effective stiffness and effective nodal forces at the interfacial boundary are evaluated by the BEM code and assembled into the FFEM system.Numerical examples confirm the effectiveness and accuracy of the method in this paper.