Reanalysis of the Extended Finite Element Method for Crack Initiation and Propagation

Abstract

The extended finite element method allows one to represent strong (cracks) and weak (holes, material interfaces) discontinuities independent of the finite element mesh through the partition of unity. This allows one to avoid costly remeshing which occurs in the vicinity of the crack tip in the traditional finite element framework when modeling crack growth. However, fatigue crack growth simulation has been computationally challenging due to the large number of simulations needed to model growth to failure. Reanalysis techniques are well developed in the areas of design and optimization for the modification of the finite element stiffness matrix to account for the addition/modification of degrees of freedom as a result of the design change. In this paper, it is observed that modeling quasi-static crack growth in the extended finite element framework involves the addition of degrees of freedom to a system of equations. Therefore, a new reanalysis algorithm based on an incremental Cholesky factorization is introduced for modeling quasi-static crack growth in the extended finite element method. This method is also used to predict the angle of crack initiation using an optimization algorithm. The examples contained within show that a 30-48% reduction in the total computational time is achievable for using the reanalysis approach for solving optimization problems or modeling quasi-static growth. It is shown that the assembly time for the stiffness matrix is insensitive to the number of elements for the proposed method.