A preconditioning method with the Generalized−α time discretization for dynamic crack propagations based on XFEM

Abstract

In this paper, we consider the efficient simulations of dynamic crack propagations based on the Extended Finite Element Method (XFEM). For the time discretization, the Generalized−α method is adopted to instead of the commonly used Newmark method in engineering, and the non physical numerical oscillations can be reduced in the Generalized−α method by choosing appropriate parameters. Moreover, in order to accelerate the convergence rate of the linear system arising from XFEM, a special crack-tip domain decomposition preconditioning method is developed, in which the computational domain is decomposed into regular subdomains and crack tip subdomains. To construct the Schwarz preconditioners, the subproblems are solved exactly in the crack tip subdomains and inexactly in the regular subdomains by an incomplete LU factorization. When cracks propagate, only the subdomains around the crack tips are updated, and all the other regular subdomains remain unchanged, which can save the computational cost significantly. The numerical experiments verify that the proposed preconditioning algorithm works well for the simulations of dynamic crack propagations.