A multi-dimensional space method for dynamic cracks problems using implicit time scheme in the framework of the extended finite element method

The extended finite element method (XFEM) has been widely used to investigate the moving cracks without any remeshing. The degrees of freedom (DOFs) of nodes around crack surfaces and crack tips are added to represent the discontinuous fields and asymptotic fields. However, the XFEM encounters some challenges in dynamic problems because the total number of DOFs increases with the crack growth, which leads to difficulty in iteration algorithm of time integration. In this paper, based on the XFEM, a multi-dimensional space method is developed to simulate the crack growth under dynamic loads. Every node, not limited to the nodes around crack tips and crack surfaces, has 12 DOFs in the domain containing one crack (12n DOFs in the domain containing n cracks). The generalized boundary condition is proposed to modify the matrices of stiffness, mass and damping. The modified matrices of stiffness, mass and damping are highly sparse, the additional central processing unit time is little and the use of internal memory does not increase compared with the traditional XFEM. Moreover, the implicit Newmark time integration scheme is introduced to solve the dynamic problems of XFEM. Compared with the other time integration schemes, the time step of the present method is much longer. The convergence property of the present method is demonstrated in theoretical and numerical studies. In order to verify the robustness and the precision of the present method, three numerical cases are presented.