A preconditioning method with auxiliary crack tip subproblems for dynamic crack propagation based on XFEM

Abstract

In this paper, a domain decomposition based preconditioning method is developed to accelerate the Krylov subspace method for solving the linear system arising from the extended finite element discretization of the dynamic crack problem. Based on the observation that the crack tip area has a significant impact on the convergence of the iterative method while the other mesh points do not, the finite element mesh is partitioned into two types: the regular subdomains and the crack tip subdomains. To construct the additive Schwarz preconditioner, the global matrix is partitioned into submatrices which are solved exactly in the crack tip subdomains by the LU factorization and inexactly in the regular subdomains by an incomplete LU factorization. As the crack propagates, we develop a scheme to update the subdomain problems without resolving of them, in which, in order to save the computational cost, only the subdomains around the crack are updated, and all the other regular subdomains remain unchanged. To further speed up the Krylov subspace method, an auxiliary subproblem including crack tips of the previous and current time steps is constructed and solved to provide a better initial guess. We carefully studied the properties of the linear system and the performance of the proposed algorithm. The numerical experiments indicate that the proposed method works well for the simulation of dynamic crack propagation with multiple crack tips.